TI-92 Geometry Tourjwilson.coe.uga.edu/emt668/emat6680.f99/glazer/92... · Web viewTI-92 Geometry...

TI-92 Geometry Tour

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The following materials were produced by the Glenbrook South High School mathematics department and may not be reproduced without permission (phone: 847-486-4683 or e-mail: [email protected]).

The following tutorials and reference material are to help you learn geometry through a dynamic environment. This means you can interact and manipulate with geometric objects without having to redraw them. The geometry program on the TI-92 calculator will accompany your textbook, UCSMP Geometry, and will be used throughout the school year to help you conjecture and learn mathematical ideas and principles.

The lessons in this guide are established through a building process. In each subsequent lesson, you will be asked to recall how you created an object from a previous activity. Therefore, it might be difficult to jump around between lessons. If you work through all of the lessons, and forget how to create something, use the Index in the back to help guide you to the exploration where it was first explained. Keep this handout for future reference.

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TABLE OF CONTENTS

EXPLORATION

0-10-20-30-40-50-60-70-80-9

0-10

1-3 1-5

2-4 2-72-8

3-1 3-23-3 3-6 3-7 3-8

3-8P

4-14-24-34-44-54-64-7

5-25-45-5

5-5P5-7

6-1

TITLE

Hey, my screen isn't dark enough (too light)!................................Oops! I pressed the wrong key......................................................Create a new folder.......................................................................Create a new document.................................................................Create, label, and move points......................................................Correct an error or clear the screen................................................Create two intersecting lines.........................................................Create, measure, and change a circle.............................................Create, measure, and change an angle...........................................Create, measure, and change a triangle..........................................

Equation of lines...........................................................................Drawing in perspective..................................................................

Midpoints......................................................................................Triangle inequality........................................................................Conjectures ................................................................

Angle bisector...............................................................................Rotations ...................................................................Vertical angles..............................................................................Slope and parallel lines.................................................................Perpendicular lines........................................................................Perpendicular bisector...................................................................Find the hidden treasure................................................................

Reflecting points...........................................................................Reflecting figures in a coordinate plane........................................Reflections and minimum distance................................................Translations: Composing reflections over parallel lines.................Rotations: Composing reflections over intersecting lines..............Translations and vectors................................................................Glide reflections............................................................................

Congruence and equality...............................................................Alternate interior angles................................................................Perpendicular bisector theorem.....................................................Capture the flag.............................................................................Sum of angles in a polygon...........................................................

Reflection-symmetric figures........................................................

PAGE(S)

5555-66-77-889-101011

12-1313-14

1515-1616-17

18192020-212222-2323-24

242526-2727-2828-2929-3030

31313232-3333-34

34-35p. 3


6-26-2P

6-36-46-5

6-M6-66-7

Isosceles and equilateral triangles..................................................Shark attack..................................................................................Constructing parallelograms..........................................................Constructing a kite........................................................................Constructing a trapezoid................................................................Mystery quadrilaterals...................................................................Rotation symmetry........................................................................Regular polygons..........................................................................

35-3636-3737-3838-3939-4040-4243-4444-45

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TABLE OF CONTENTS (CONTINUED)EXPLORATION

7-27-57-67-7

7-7P7-9

8-18-2

8-2P 8-3 8-4

8-4P8-5 8-6 8-7 8-8

8-8P

9-5

10-6

11-6 11-7 11-8

11-8P

12-1 12-2 12-3 12-4 12-5 12-6

13-113-2

13-2P13-3

TITLE

Congruent triangles.......................................................................SsA condition and HL congruence................................................Tessellations..................................................................................Properties of parallelograms..........................................................Bouncing off the walls..................................................................Exterior angles..............................................................................

Perimeter of a regular polygon......................................................Area of a rectangle........................................................................Optimal quadrilaterals...................................................................Areas of irregular polygons...........................................................Area of a triangle..........................................................................Triangle in a rectangle...................................................................Area of trapezoids.........................................................................Pythagorean Theorem...................................................................Circumference and arc length........................................................Area of a circle through data analysis............................................Area of tangent circles..................................................................

Famous paths in geometry...............................................

Creating a toolbar to reference formulas.......................................

Distance formula...........................................................................Equations of circles.......................................................................Means and midpoints....................................................................Midpoints and areas......................................................................

The transformation sk....................................................................Size changes..................................................................................Properties of size changes.............................................................Proportions....................................................................................Similarity......................................................................................The fundamental theorem of similarity.........................................

The SSS similarity theorem...........................................................The AA and SAS similarity theorems...........................................Triangles in a trapezoid.................................................................The side splitting theorem.............................................................

Index.............................................................................................Variable Index...............................................................................

PAGE(S)

45-4747-4849-50505152

5353-5454-5555-5656-5858-5959-6161-6262-6364-6666-67

68-69

70-71

72-7373-7575-7677-78

78-7980818282-8384

84-8585-8686-8787

88-9192-93

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INTRODUCTION TO THE TI-92

Objective: In this activity, you will practice some of the basic functions of the geometry on the TI-92 calculator, including use of the Pointer, Point, Line, Measure, and Construct toolbars.

EXPLORATION 0-1. HEY, MY SCREEN ISN'T DARK ENOUGH (TOO LIGHT)!Turn the calculator . and will darken your screen one notch if necessary. and will lighten your screen one notch if necessary. Repeat until you reach the desired contrast. And once you’ve gotten your screen adjusted the way you like it, you can reduce glare by snapping the lid to the calculator. You may find it easier not to do this, however, and that’s fine, too.

EXPLORATION 0-2. OOPS! I PRESSED THE WRONG KEYThere are a lot of keys in this small hand-held computer (large calculator). Therefore, it is natural to press the wrong key at the worst possible moment (that's called Murphy's Law). When you press the wrong key, press the button to get you back to where you started. Also, press when you want to return to the pointer (cross). Get in the habit of pressing the escape key when you’re done with a certain feature.

EXPLORATION 0-3. CREATE A NEW FOLDERFirst, you need to create a folder to save all of the work you do in this course.Press , and 5:Create Folder. Type your name in the Folder: box and press twice. A folder with your name should now appear in the directory. You should not need to do this again for the remainder of the year.

EXPLORATION 0-4. CREATE A NEW GEOMETRY DOCUMENTStart the geometry program by pressing , and choose 8:Geometry » 3:New... to begin. When you select Geometry and New you can either press the corresponding number (8 and then 3) or you can arrow down using the arrow pad (blue wheel). Use the arrow pad to arrow right and arrow down to your name and press to change the Folder: to your name. Arrow down (do not press enter) and type points in the Variable: box and press twice. You should now see a pointer (cross) at the center of the screen and a bunch of tools to represent menus at the top. Use this process each time you want to create a new geometry file. You may have to refer back to this page until you get comfortable with it.

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Now take a minute to explore the different function keys, or F-keys, like F1, F2, . . . Don’t actually select any particular function. Just select each function key and see what options are in that menu.

EXPLORATION 0-5. CREATE, LABEL, AND MOVE POINTS.Press and choose 1:Point. Note the cross has changed to a pencil, meaning that you are ready to create points on the screen. Use the arrow pad to move the pencil to any location on the screen and press . Immediately type A to name the point. To make capital letters (which is customary for points) just hit

It is best to immediately type the letter before another point is created. If you keep creating points, it’s much more difficult to label the points. Use the arrow pad to move the pencil away from the point so you can see it on the screen. Press again to make another point and immediately type B to name it. Repeat this process until you have made five points named A through E.

If you forget to label or mislabeled a point, first press to get out of the point making function. Your calculator can’t read your find and will assume you’re making more points until you tell it otherwise. After hitting you should see the pointer (cross), move the arrow pad over to the point until the screen reads THIS POINT, press and choose 4:Label, press , use the left arrow key (2 keys to right of space bar) to delete the old label (if there was one), type in or replace the label, and press .

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Press if the pointer (cross) is not on the screen. Use the arrow pad and move on top of point c until the screen reads THIS POINT. Hold down the with your left thumb and move point c around the screen by using the arrow pad with your right thumb. Note: The and arrow pad (blue wheel) together function like a mouse. Just like when you click and drag something on the computer with a mouse you must hold the button down, on the TI-92 you must hold the button down while you use the arrow pad for the movement.

Now let's just reposition (or drag) the label. Use the arrow pad and move on top of the label for point c until the screen reads THIS LABEL. Hold down the with your left thumb and move the label for point c with the arrow pad .

Move the points and labels around until your screen matches the one shown to the right. When you move a point the label follows.

EXPLORATION 0-6. CLEAR SCREEN/DRAW SEGMENT/CORRECT AN ERRORLet's start over with a fresh screen by pressing and choosing 8: Clear All. The calculator will ask you to press enter to confirm this request.

Next draw a segment anywhere on the screen by pressingand 5: Segment. Notice that the icon in the "toolbar" across the top of the screen for F2 has changed from a point to a segment. Use the arrow pad to move to where you want the segment to start, press -- don't forget to type a letter immediately after so you don't have to use the label command. Then repeat for the other endpoint and type a label.

The following information is needed when you make an error. If you press the wrong key, press to clear the command. Let's practice deleting things. First press to get back to the pointer and to tell the calculator that you're no longer interested in making segments. Move the key pad on top of the object until it says THIS SEGMENT (it won't say this if you point at the endpoints -- you need to move inbetween the points somewhere), press , press and choose 7:Delete (or press the LEFT ARROW). Notice that the

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points are still on the screen. You could delete each separately, but instead select a point and then hold down the shift key (the UP ARROW).

EXPLORATION 0-7. CREATE TWO INTERSECTING LINESCREATE A NEW DOCUMENT (even though your screen is blank -- you need to practice) with the variable lines. Refer to Step 4 if you forgot. Get in the habit of 1. referring back when you forget something, then 2. ask a group member, then 3. ask the teacher if needed and I'll be glad to help. A line can be created with two points. Select and choose 4:Line. Move the pencil to the lower-right side of the screen and press . Label this point A. Move the pencil to the upper-left side of the screen so that the line is on a diagonal and press . A line like this, which is neither horizontal nor vertical, is called oblique. Create another line so that the two lines create a large X as shown to the lower-left. Label the point on the second line B.

Press and choose 3:Intersection Point. Move the pencil to the intersection point until the message POINT AT THIS INTERSECTION appears. Press , label this point C, and to return to the pointer. Move the cross to the upper-right hand portion of the screen until the screen reads THIS LINE. Press and choose 2:Point on Object, press , label this new point D (be careful not to label the line), then . Your screen should match the picture shown above to the right.

EXPLORATION 0-8. CREATE, MEASURE, AND CHANGE A CIRCLECREATE A NEW DOCUMENT with the variable circ.

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A circle is formed by selecting a center and a point on the circle. The radius is the distance from the center of the circle to any point on the circle. Press and choose 1:Circle, press to mark the circle's center and type C for center, and move the pencil outward to establish the size of the circle's radius. Make a circle large enough to cover one-fourth of the screen and press to finish the circle. Construct a radius by pressing & choosing 5:Segment, move the pencil on the circle's center, pressand then drag the segment to a point on the circle until the screen reads ON THIS CIRCLE, and press and type A. Press to return to the pointer.

Measure the radius of the circle by pressing and choosing 1:Distance & Length, move the pencil to the middle of the segment until the screen reads LENGTH OF THIS SEGMENT, and press . Press to return to the pointer, move the pencil on top of the radius measurement, press to select (dotted), and to delete the measurement.Another method to measure the length of a segment is to find the distance between the endpoints by moving the pencil to one endpoint, pressing , moving to the other endpoint, and pressing . Press to return to the pointer, drag the measurement to the upper right-hand portion of the screen, press and choose 5:Comment, drag the pencil to the left of the measurement, press , type radius= , hit , and then drag the comment as needed. Consult the diagram below.The circumference of the circle is the length around the circle. Measure the circumference by pressing and choosing 1:Distance & Length, move the pencil to the circle until the screen reads CIRCUMFERENCE OF THIS CIRCLE, and press . Type a comment (abbreviate with circum if you wish) for the measurement and drag it underneath the radius measurement .

The area of the circle is the amount of space inside the circle. Measure the area of the circle by pressing and choosing 2:Area, move the pencil to the circle until the screen reads THIS CIRCLE, and press . Comment (area=) and drag this measurement underneath the other measurements. Drag the circle to the left-hand side of the screen by moving its center as shown below to the left.

You can change the size of the circle by moving the pointer on top of the circle, pressing , and dragging with the toward or away from the circle's center. Modify the measurements on your screen until they match the picture shown to the right above (within 0.1 cm and 0.3 cm2).

EXPLORATION 0-9. CREATE, MEASURE, AND CHANGE AN ANGLE

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CREATE A NEW DOCUMENT with the variable angles. An angle can be created with two rays. Press and choose 6:Ray. Move the pencil to the left-hand-center part of the screen and press . Label this vertex point B and direct the ray to the upper right-hand portion of the screen and press . Move the pencil on top of point B until the screen reads THIS POINT, press to make another ray, and direct the second ray to the lower right-hand portion of the screen and press . Create point A on the top ray and point C on the bottom ray by using and 2:Point on Object as shown in the diagram below to the left.

Use the to move points A and C close to the vertex of the angle. An angle can be measured if a point on one ray, the vertex, and a point on the other ray are selected. Press and choose 3:Angle. Move the pencil over point A and press , over point B and press , over point C and press . The measure of <ABC should appear on the screen. Press to return to the pointer and move the pencil along ray BC until the screen reads THIS RAY. Hold down the and use the key pad to change the angle until it almost equals 65˚ (within 0.5˚). If necessary, move the angle measurement inside the angle, and the points on the angle closer to the vertex so they remain on the screen as shown in the picture to the top-right.

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EXPLORATION 0-10. CREATE, MEASURE, AND CHANGE A TRIANGLECREATE A NEW DOCUMENT with the variable triangle. Press and choose 3:Triangle. Press and label the first point a, drag the pencil and repeat the process until you have created ∆abc. Use the procedures in Steps 8 and 9 to measure the triangle's perimeter, area, segment lengths, and angle measurements. Organize the measurements as shown in the image below to the left.

It is possible to make calculations on the screen to explore geometric patterns. Press and choose 6:Calculate. A calculation strip with a cursor will appear at the bottom of the screen. Using the arrow pad (blue wheel) arrow up until one of the side length measurements is highlighted, press and an "a" should appear on the calculation strip (the number has been stored in what the calculator calls "a"),type a since we want to add it to the other sides, highlight another side length measurement by arrowing up, press and a "b" should appear on the calculation strip, , highlight the last side length measurement, press and a "c" should appear on the calculation strip. Press to execute the calculation and a value next to an R: should be on the screen as shown above to the right.

Press to return to the pointer and drag one of the vertices of the triangle around the screen and observe the changes in the numbers and the similarity between this number and one of your measurements. Explain what you have found below.

Find the sum of the angles in a triangle. Drag the vertices around to modify the angle measurements. Explain what happens to this sum. Why do you think this is happening? Explain below.

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EXPLORATION 1-3. EQUATION OF LINES.

Objective: Distinguish the equations of vertical, horizontal, and oblique lines.

1. CREATE A NEW DOCUMENT with the variable eqline.

2. Create a coordinate planePress and choose 9:Format, change the Coordinate Axes to RECTANGULAR and the Grid to ON, press to save, move the cross on top of the unit on the x-axis (probably 0.5) until the screen reads THIS UNIT, hold the , and drag the unit until it changes to 1 as shown in the picture below to the left.

3. Place points at (3,2) and (-1,-1). Create a line which passes thru both of these points. Make sure the screen reads THRU THIS POINT before pressing , as shown in the diagram above to the right.

4. Find the equation of the line in slope-intercept formPress and choose 5:Equation & Coordinates, move the pencil on top of the line until the screen reads EQUATION OF THIS LINE, press and the equation should appear next to the pencil in slope-intercept form. Write the equation of the line below, and state the slope and y-intercept.

EQUATION: ______________________ SLOPE: ________ Y-INTERCEPT: ___________

5. Press to return to the pointer. Drag the point (3,2) to the point (3,-1). Write the equation of the line below, and state the slope and y-intercept.


6. Drag the point (3,-1) to the point (-1,2). Write the equation of the line below, and state the slope and y-intercept.


7. Explain below how you can distinguish vertical, horizontal, and oblique lines by their equation.

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8. Use your explanation in step 7 to determine whether the following equations represent vertical, horizontal, or oblique lines. Use your calculator only to CHECK your answer.

Equation Your Answer Passes Thru (to check on calculator)a. x = 2 _____________ (2,-1), (2,2)

b. y = -3 _____________ (0,-3), (-4,-3)

c. y = x - _____________ (-1,-1), (1,0)

EXPLORATION 1-5. DRAWING IN PERSPECTIVE

Objective: In this activity, you will create a box in perspective where you can manipulative the vanishing point to see different viewpoints. Use page 29 of your text as a reference.

1. CREATE A NEW DOCUMENT with the variable perspect.

2. Create a square Press and choosing 5:Regular Polygon, move the pencil near the center of the screen, press to locate the center of the square, move the pencil about 2 cm away from the center and press , move the pencil away from the point on the dotted circle until the screen reads 4 (sided polygon) and press as shown to the right. A square should now appear on your screen.

3. Place a point on your screen to represent the vanishing point as shown in page 29 on your text.

4. Connect segments from the vanishing point to three of the vertices of the square as shown in the figure at the right. Make sure the calculator reads THIS POINT before pressing .

5. Create parallel lines to the sides of the square which connect to the vanishing point.

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Press and choose 2: Parallel Line, move the pencil to the right-hand side of the square until the screen reads PARALLEL TO THIS SIDE OF THE POLYGON, press , move the pencil on top of the lowest segment until the screen reads ON THIS SEGMENT and press . The parallel line should now appear on the screen. Create a point of intersection between the parallel line and the middle perspective line. Now create another parallel line thru the point of intersection and parallel to the top segment of the square as shown in the diagram to the right. Create an intersection point between the new parallel line and the left-most perspective segment.

6. Create five overlapping segments which create the three dimensional perspective of the box.

7. Hide everything except the box and vanishing point. Press and choose 1:Hide/Show, move the pointer on each object you wish to hide and press . The object(s) you are hiding should become dotted. When you are finished hiding, press to return to the pointer. If you make a mistake, undo, and try again. Your screen should resemble the picture to the right.

8. From the current perspective, you can view three faces of the box. Move the vanishing point around the screen and observe the changing perspectives of your box.

Where is the vanishing point when the top face is not showing?

Where is the vanishing point when the side face is not showing?

Where is the vanishing point when the top and side face are not showing?

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EXPLORATION 2-4. MIDPOINTSObjective: Determine how a midpoint relates to and affects a line segment.1. CREATE A NEW DOCUMENT with the variable midpoint. 2. Place an oblique segment across 3/4 of the screen. Label the endpoints d and g.3. Create a midpoint on a segmentPress and choose 3:Midpoint, place the pencil on top of the segment until the screen reads midpoint of this segment, press and label the midpoint o.

4. Measure the length of segments do, og, and dg. Label the measurements and organize them on the screen as shown to the right.5. Drag point d around the screen and observe the changes in your measurements. Make two conjectures below about the effect of a midpoint on a segment.6. Clear everything from the screen by pressing and 8:Clear All. Press to confirm.7. Create a large circle in the center of the screen. Label the center u. Construct a line which passes thru the circle and the center of the circle. Make a point of intersection with the circle and the line. Create an overlapping segment connecting the two points shared by the line and the circle. This segment which has endpoints on the circle and passes thru the center of the circle is called the diameter of the circle. Label the diameter endpoints c and b. Hide the line so your diagram is similar to the one shown to the right.8. Measure the radii (segments cu and ub). Drag and modify the size of the circle and observe changes in your measurements. Explain below how the center, radii, and diameter of the circle relate to the concept discussed earlier this exploration.

EXPLORATION 2-7. TRIANGLE INEQUALITY

Objective: Determine how the lengths of the side of a triangle relate to each other.

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1. CREATE A NEW DOCUMENT with the variable inequal.

2. Create a large triangle (, 3) which covers about one-half of the screen. (Remember to type the label after each time you press enter to make each of the three vertices.) Measure the length of each of the sides of the triangle and label each of the measurements.

3. Calculate the og + gp, og + op, op + gp, label and organize the calculations as show to the right.4. Compare the sum of two sides of the triangle to the third side. Record and organize the information below.

sum of two sides og + gp = og + op = op + gp = third side op = gp = og =

How do the sums compare to the 3rd side? Which one is greater?

Move point o close to segment gp. Record the data below.



Move point o on top of segment gp so the triangle turns into a segment. Record the data below.



In a triangle, make a conjecture about the information above using an if...then... statement.

EXPLORATION 2-8. CONJECTURES

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Objective: Determine the validity of conjectures

1. CREATE A NEW DOCUMENT with the variable conject.

2. Create a large triangle. Construct the midpoint of each side and connect them with three segments as shown.

3. Test the following conjecture: If the midpoints of two sides of a triangle are joined, the segment is parallel to the third side.At this point, do you feel this conjecture is true? How sure are you?

To discern with more certainty whether this is true, press and choose 8: Check Propery. Arrow right to choose Parallel from the sub-menu. Let's check if

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ED

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Now drag around one of the vertices of the original triangle by pressing , moving to a point, and pressing the hand tool. Do you find that this conjecture is true in general?

4. Test the following conjecture: Area(∆ABC) = 4 • Area(∆DEF) or, in other words, the ratio of their two areas is 4. At this point, do you feel this conjecture is true? How sure are you?

Have the calculator find the areas of the triangles. However, recall that we made ∆DEF by joining three segments. The calculator cannot find the area of a region that was not constructed as a polygon. Therefore, we must make an overlapping triangle on top of the existing segments. Create ∆DEF and then you will be able to find its area. Drag the measurements to the upper right hand portion of the screen.

Next we will use the 6: Calculate tool under the menu to divide the larger area of ∆ABC by the smaller area of ∆DEF. If you've forgotten how to do this, refer back to Exploration 0-10. Remember that you have an index! I won't usually tell you where to refer -- you're expected to use the index or ask a group member.In your calculation strip across the bottom you should see a/b (you only type the which is on the calculator, the 92 does the letters when you highlight a number and press enter).

Press since we no longer wish to calculate. Get in the habit of pressing escape when you're done with something! Now drag around one of the vertices of the original triangle. Be sure to move around in all directions. Do you find that this conjecture is true in general?

EXPLORATION 3-1. ANGLE BISECTORObjective: Determine how an angle bisector relates to and affects an angle.1. CREATE A NEW DOCUMENT with the variable angbis. 2. Create an angle and label it so it reads < TAL. (the symbol is in the catalog, so it is easier to just use the "less than" symbol for angles. The < is typed by pressing and .) Measure and label (comment) the angle. See EXPLORATION 0-9 if you forgot.

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3. Create an angle bisectorPress and choose 5:Angle Bisector, move the pencil on top of each point IN ORDER, and press after selecting each point. The angle bisector should divide TAL into two smaller angles. Place a point on the angle bisector and label it C as shown to the right.

4. Measure and label the two smaller angles, TAC and CAL.5. Drag one of the rays around the screen so the angles change measurement, and observe the changes in your measurements. Make two conjectures below about the effect of an angle bisector on an angle.

6. Construct a triangle and the angle bisectors of all of the angles. Complete the following conjecture: If the angles of a triangle are bisected, then ...

7. What side of a triangle is closest to the intersection of the angle bisectors? Explain how you know.

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EXPLORATION 3-2. ROTATIONSObjective: Determine how magnitude and sign of a rotation affects the preimage1. CREATE A NEW DOCUMENT with the variable rotate.

2. Create a small, pointy triangle as illustrated. Create a point, O, near the triangle. To create the numerical value of the rotation, use and choose 6: Numerical Edit. Press to get the edit box and type 90. Press .

To rotate the triangle 90° in the positive direction, select and 2: Rotation. Select the triangle to rotate, then point O as the center of rotation, and finally select the 90, and presto!

3. With a 90° rotation, how much did the triangle turn? Use answers such as quarter-turn, half-turn, three-quarters turn, etc.

Make a sketch of the preimage and image in the box below. Label the preimage Start and label the image 90 in your sketch.

SKETCHES . 4. Undo the 90° rotation by hitting and choose D:Undo or press and . Change the 90 to 180 by arrowing to the number, pressing twice (once to select and once to get the edit box to appear) and then

Sketch the resulting image triangle in the same drawing as the 90° rotation. Label this triangle 180°.

Repeat these steps for a 270°, 360°, and 450° rotation. Answer the questions below and make a sketch.

With a 270° rotation, how much did the triangle turn?



4. What happens with a negative rotation? Try a numerical edit of -45°. Be sure you use the gray key for the negative. Describe the difference between negative and positive rotations.

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EXPLORATION 3-3. VERTICAL ANGLES

Objective: Determine the relationship between two vertical angles.

1. CREATE A NEW DOCUMENT with the variable vertang.

2. Create two lines intersecting near the middle of the screen. Find a point of intersection, create and label points on the lines so they correspond with the diagram to the right.

3. Measure vertical angles DAM and GAO. Repeat this procedure with vertical angles GAM and OAD. Drag around the lines so the angle measurement change.4. Make a conjecture below about your findings in step 3.

EXPLORATION 3-6. SLOPE AND PARALLEL LINES.

Objective: Verify the slope formula and determine some properties which relate to parallel lines.

1. CREATE A NEW DOCUMENT with the variable parallel.

2. Use the Format tool in to turn on the coordinate axes to a RECTANGULAR grid. Turn the grid to ON. Press to save these changes. Change the unit label 0.5on the axes to equal 1 by first getting the prompt THIS UNIT. This is a little tricky. Move around until you get it. You don't want THESE AXES but the calculator will try to get you to bite. Don't. Once it says THIS UNIT (as shown below to left), slowly drag it left to get a grid like the one to the right.

3. Create a line passing thru (-1,-1), and (1,0). Label the line b after pressing on the second point, as shown above to the right.

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4. Measure the slope of the linePress and choose 4:Slope, move the pencil on top of one line until the screen reads THIS LINE, press . Label the slope with slope b=

5. Use the two ordered pairs and the slope formula to verify that the slope is 0.50. Show your work below.

6. Create a parallel linePress and choose 2:Parallel Line, move the pencil on the line until it reads PARALLEL TO THIS LINE, press , move the pencil away to where you want your new line and press , label the new line a.

7. Measure and label the slope of line a. Drag the original line (not at a defined point) changing the tilt (slope) of it. Compare the changes in the slopes of the two lines.

8. Make an IF...THEN...conjecture below about your findings in step 7.

Note: Try dragging the second line that was created to be parallel to the original. What is different? . . . The slope must remain the same as the original line because it was created dependent upon it and the calculator remembers how things are constructed. Therefore, if you wish to change things around, you must drag the original, not the dependent.

9. Use the Format tool and turn off the Coordinate Axes and Grid (to make the screen neater for measurements).

10. Create a transversal, a line which crosses both of the parallel lines. Measure one pair of corresponding angles as shown in the diagram to the right. Note: You can create points on the lines as you measure the angles, including the intersection points. Move the measurements inside the angle as shown to the right. Drag line b around and observe the change in the angle measurements of the corresponding angles.


12. How many more pairs of corresponding angles exist in the diagram? Without measuring, explain below why each pair should be congruent.

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EXPLORATION 3-7. PERPENDICULAR LINES.

Objective: Determine the properties which determine perpendicular lines.

1. CREATE A NEW DOCUMENT with the variable perpline.

2. Create an oblique line, labeled n.

3. Create a perpendicular linePress and choose 1:Perpendicular Line, move the pencil on top of line n, and press twice (once to identify the line and another to identify the location).

4. Measure one of the angles created by the intersecting lines. Drag line n around and observe changes in the angle measurement.


6. Measure and label the slopes of the lines. Using the calculate tool in , calculate and use a comment box to label the product of the slopes as shown to the right. Drag line n around and observe changes in the product of the slopes.


EXPLORATION 3-8. PERPENDICULAR BISECTOR.Objective: Construct and understand the properties of a perpendicular bisector using three methods.1. CREATE A NEW DOCUMENT with the variable perpbis.

2. Create a perpendicular bisectorCreate an oblique segment with endpoints labeled b and s. Press and choose 2:Perpendicular Bisector, move the pencil on top of the segment until the screen reads PERPENDICULAR BISECTOR OF THIS SEGMENT, and press . This line shown to the right is called the perpendicular bisector of segment bs.

3. Create a midpoint on segment bs labeled i. Construct a perpendicular line thru point i. Drag the endpoints of the segment around and compare this new line with the line in step 2.4. Make an IF...THEN...conjecture below about the properties of a perpendicular bisector.

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5. Clear the screen and create a small oblique segment with endpoints a and b near the center of the screen.6. Create two circles that have their centers and radius points at the endpoints of the segment as shown in the diagram below to the left. Construct a line passing thru the intersections of the circles. Label the points of intersection c and d. Create and label the point of intersection of segment ab and the line as shown in the diagram below to the right.

7. Line CD is supposed to be the perpendicular bisector of segment AB. Make calculations on the screen and drag the segment's endpoints to verify this statement. Explain your process and affirmation below.

EXPLORATION 3-8P. FIND THE HIDDEN TREASURE

Objective: Use rotations and perpendicular bisectors to locate a specific spot on a map.

1. CREATE A NEW DOCUMENT with the variable treasure.

An old map describes the location of buried treasure thus: "On the island there are only two trees, A and B, and the remains of a ship. Start at the ship and count the steps required to walk in a straight line to tree A. At the tree turn 90 degrees to the left and then walk forward the same number of steps. At the point where you stop drive a spike into the ground. Now return to the ship and walk in a straight line, counting your steps, to tree B. When you reach the tree, turn 90 degrees to the right and take the same number of steps forward, placing another spike at the point where you stop. Dig at the point exactly halfway between the spikes and you will find the treasure."

2. Explain how to construct a diagram that will locate the position of the treasure using rotations.

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3. When you get to the island you find the ship missing! Is there any way you can still get to the treasure? Explain your strategy. No, you may not dig up the entire island; you may dig only once!!!

4. Will the treasure be in the same position if the ship is in a different position? Will the treasure be in the same position if the trees are in a different position? Explain why or why not.

5. Will your strategy still work if the ship is in a different position? Will your strategy still work if the trees are in a different position? Explain why or why not.

EXPLORATION 4-1. REFLECTING POINTS.

Objective: Understand the properties of a reflection.

1. CREATE A NEW DOCUMENT with the variable reflect.

2. Place an oblique line on the screen. Place a point labeled p close to, but not on, the line.

3. Reflect point p over the linePress and choose 4:Reflection, move the pencil over point p until the screen reads REFLECT THIS POINT, press , move the pencil over the line until the screen reads WITH RESPECT TO THIS LINE, press , and label the new point p' (press , to access the catalog of terms and symbols, then move up until you find the prime symbol ', and press . It is under and above " ).

4. Move point p around the screen and describe the changes below. What if point p is on the line?

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5. Create a line segment connecting points p and p'. How is the reflecting line related to the segment pp'? Make calculations on the screen and drag the segment's endpoints to verify this statement. Explain your process and affirmation below.

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EXPLORATION 4-2. REFLECTING FIGURES IN A COORDINATE PLANE.

Objective: Understand reflection in a coordinate system.

1. CREATE A NEW DOCUMENT with the variable coordin.

2. Use the Format tool to turn on the coordinate axes to a rectangular grid. Change the unit label on the axes to equal 1 as shown below to the left.

3. Create ∆ABC with vertices on the coordinates {(-1,3),(5,2),(5,-1)} as shown on the diagram above to the right. Press each time the screen reads ON THIS POINT OF THE GRID.4. Reflect the triangle over the y-axis, label the new coordinates A', B', and C', and record the coordinate of the three points below.

A A' B B' C C' coordinates (-1,3) (5,2) (5,-1)

5. If you did not have a calculator, explain in a complete sentence below how you would find the coordinates of a figure reflected over a line.

6. Suppose the triangle is reflected over the horizontal line y=1, predict the coordinates of the vertices of the image without using the calculator.

A A' B B' C C' predicted (-1,3) (5,2) (5,-1)coordinates

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7. Undo (,) the image ∆A'B'C', draw the line which represents y=1, and reflect the triangle over the line to check your answer.EXPLORATION 4-3. REFLECTIONS AND MINIMUM DISTANCE

Objective: Use reflections to find a minimum distance along a path.

1. CREATE A NEW DOCUMENT with the variable swimmer.

2. A long pier,

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3. Hide the lines as shown in the diagram to the right. Recall that when you use the Hide/Show command, select the command from the menu, then select the object. Also remember to hit escape after pressing enter to see the line disappear. Find the three lenths needed ( The scale will be 1 cm = 1mile) and adjust the points until the measurements match those in the diagram above.

4. Make two segments from A to a point on the shore and from C to a point on the shore. Label this point B. Notice in the picture that these two segments are dotted. To do this, select and 9: Dotted. Press when done.

5. Measure and label the distance of the 1st swim (AB) and the second swim (BC). Use the Calculate tool to find the distance of the total swim, as shown to the right. Use 5: Comment under for the measurement descriptions.

6. Measure the distance from the rest stop on the beach to the intersection of the beach and the pier (point D). Drag point B along the shoreline until you find the spot on the beach/land that will allow the least amount of total swimming. Leave B at this optimal spot and record the total swimming distance and the distance from the shorter pier (along the beach).

Minimum total swimming distance (AB + BC) =

Distance from the shorter pier on the beach (BD) =

7. Now you will see the powerful use of a reflection in this application.

Reflect point C over the beach, creating C'. Create segment AC' as shown to the right, and observe the results.

Note: The position of point B on your screen should be different from the screen shown to the right.

8. Explain below how you would solve this problem without a calculator. Also explain why this method helps you find the correct answer.

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EXPLORATION 4-4. TRANSLATIONS: COMPOSING REFLECTIONS OVER PARALLEL LINES.

Objective: Understand composition of reflections over parallel lines.

1. CREATE A NEW DOCUMENT with the variable compref.

2. Create a triangle to the left of two parallel lines as shown to the right.

3. Construct a perpendicular line. Create an intersection point where the lines intersect. (One should already be there) Find the distrance between the two points along the perpendicular. Move the lines so they are approximately 1.5 cm apart. Wewill call this distance d1. (You don't need to label this)

d1

4. Reflect the triangle over the line closest to the triangle, and reflect the image over the next line as shown in the diagram to the left.

5. Construct a line from the vertex of the original triangle perpendicular to one of the parallel lines as shown in the diagram above to the right. Measure the distance between the same vertex in the original triangle to its corresponding vertex of the last triangle (under two reflections). This distance is shown as d2 in the diagram to the right.

Drag the original line around and modify the distance between the reflecting lines to compare the two measurements d1 and d2.

d1

d2

6. Complete the following conjecture: If a figure is reflected twice over parallel lines, then the final image will be a t _ _ _ _ _ _ _ _ _ _, or s _ _ _ _, of _ _ _ _ _ the distance between the parallel lines.

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EXPLORATION 4-5. ROTATIONS: COMPOSING REFLECTIONS OVER INTERSECTING LINES.

Objective: Understand composition of reflections over intersecting lines.

1. CREATE A NEW DOCUMENT with the variable compref2.

2. Create ∆PAL to the left of two intersecting lines which intersect at T as shown below to the left.

3. Reflect the triangle over the closest line. Reflect the image over the next line and label ∆P"A"L". Hide the intermediate image and its vertices as shown in the diagram above to the right.

4. Measure and label the angle between the intersecting lines. Measure and label < ATA", the angle between the original triangle and its second image. Drag the lines to modify the angle sizes and compare the two measurements.5. Complete the following conjecture: If a figure is reflected twice over intersecting lines, then the final image will be a clockwise r _ _ _ _ _ _ _ of ____________ the angle between the intersecting lines.6. Now you will begin confirming your conjecture. Select and choose 9:Dotted, place the pencil over the image triangle and press . 7. The calculator rotates counterclockwise, so you need to rotate by a negative angle. Use the calculate feature and multiply < ATA" by -1. This value will be used in the next step.

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8. Rotate an object about a point with a specified anglePress and choose 2:Rotation, move the pencil over ∆PAL until the screen reads ROTATE THIS TRIANGLE, press , move the pencil over point T until the screen reads AROUND THIS POINT, press , move the pencil over the negative angle calculation until the screen reads USING THIS ANGLE, and press . The rotation should place the new image on top of the dotted image, verifying your conjecture in step 6. Not convinced yet? Undo your rotation steps by pressing and . Notice the dotted image reappears!

EXPLORATION 4-6. TRANSLATIONS AND VECTORSObjective: Use vectors to translate objects on the coordinate plane.

1. CREATE A NEW DOCUMENT with the variable vector.

2. Set up the coordinate plane as in EXPLORATION 4-2.

3. Construct ∆CHI with vertices {(-6,-1),(-5,-3),(-4,-3)} and a vector (press and choose 7:Vector) with initial point (0,0) and terminal point (4,2) as shown to the right.4. Translate the triangle by the marked vectorPress and choose 1:Translation, move the pencil on top of the triangle until the screen reads TRANSLATE THIS TRIANGLE, press , move the pencil on top of the vector until the screen reads BY THIS VECTOR, and press . Label the vertices of ∆C'H'I'.5. Move the terminal point of the vector around and observe changes in the image under the translation. Use a complete sentence to explain below how a vector with initial point (0,0) and terminal point (x,y) translates the triangle.

6. There are an infinite number of vectors that will translate ∆CHI. The initial point can start anywhere, not necessarily at the origin. Suppose the vertices of ∆CHI are translated to {(-7,2),(-6,0),(-5,0)} under a translation. Find the initial and terminal points of three vectors which make this possible. Record your points below.

vector 1 vector 2 vector 3initial pointterminal pointState in terms of # of units right/left and # of units up/down the translation of the triangle.

How do the coordinates for your vectors above demonstrate this? Be specific as to x- and y-coordinates.

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7. Astro states that he used a vector to translate the vertices of ∆CHI to {(-2,-3),(-1,-5),(1,-4)}. Can you find a vector that Astro could have used to make this translation? If so, what is it? If not, explain why not.

EXPLORATION 4-7. GLIDE REFLECTIONS

Objective: Use translations and reflections to create footsteps.

1. CREATE A NEW DOCUMENT with the variable glideref.

2 Draw an oblique line thru the center of the screen and a small dinosaur footprint (3 toes) above the line as shown below to the left. Use and choose 4:Polygon to create the foot, and hide the vertices.

3. Create a small vector that moves in the direction of the line. Reflect the foot over the line and translate the image by the vector as shown in the diagram above to the right.

4. Create a macro to create the glide reflectionA macro allows you to perform multiple steps at once. This macro will perform the glide reflection, the reflection and translation without showing the intermittent image. Press and choose 6:Macro Construction » 2:Initial Objects, select the first footprint (polygon), select the vector, and select the line in the specified order until they are all dotted. Press and choose 6:Macro Construction » 3:Final Objects, and select the last footprint until it is dotted. Press and choose 6:Macro Construction » 4:Define Macro, save the file name and variable as steps in your folder. Now execute the macro by pressing and choosing 6:Macro Construction » 1:Execute Macro. Press when the name steps appears on the screen, select the footprint, select the vector, and select the line in the specified order. A new footprint under the glide reflection should appear on the screen after a short while.

Repeat the procedure until a series of footprints cover the screen as shown to the right.

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EXPLORATION 5-2. CONGRUENCE AND EQUALITY

Objective: Compare measurements of a pre-image and its image under an isometry.

1. CREATE A NEW DOCUMENT with the variable congru.

2. Create a four-sided polygon MATH (using the menu) and a point on the screen as shown to the right. Measure and label each of the angles and sides of the polygon.

3. Your teacher will instruct you to perform one of the following isometries: translation, rotation, reflection, or glide reflection.

You may need to move the polygon MATH higher on the screen so its image is completely visible.

4. Label the image polygon M'A'T'H'. Measure the angles and segments of the image. Drag around any of the vertices of the original figure MATH and observe the similarities after changes in measurements between the two figures.

5. Make an IF...THEN... conjecture below based on your findings in step four.

EXPLORATION 5-4. ALTERNATE INTERIOR ANGLES

Objective: Find the relationship between alternate interior angles in a diagram.

1. CREATE A NEW DOCUMENT with the variable altint.

2. Create a triangle. Construct a line parallel to one side of the triangle, passing through the opposite vertex as shown in the diagram to the right. Measure one pair of alternate interior angles.

3. Drag one vertex of the triangle around so the angle measurements change. Observe any relationships between the alternate interior angles.

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4. Measure the second pair of alternate interior angles. Drag around a vertex of the triangle and notice any similarities or differences from your result in step 3.

5. Make an IF...THEN... conjecture below based on your findings in steps 3 and 4.

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EXPLORATION 5-5. PERPENDICULAR BISECTOR THEOREM

Objective: Find the relationship between a segment and any point on its perpendicular bisector

1. CREATE A NEW DOCUMENT with the variable ptperp.

2. Construct a segment, its perpendicular bisector, and two points on the perpendicular bisector (one above one below the segment) as shown in the diagram below to the left.

3. Measure the distance from the points on the perpendicular bisector to the endpoints of the segment as shown above in the diagram to the right. You do not need to create the dotted segments; they are shown to help you visualize the distances.

4. Drag the points along the perpendicular bisector and observe changes in the measurements.

5. Write an IF...THEN... statement below describing your findings in step 4.

EXPLORATION 5-5P. CAPTURE THE FLAG

Objective: Determine how perpendicular bisectors can be used to create a fair game.

1. CREATE A NEW DOCUMENT with the variable capture.

Three teams, A, B, and C, each start from a location on a field. Their goal is to grab the flag located in aknown location on the field. If the game is fair, then each team has to run the same distance to the flag.

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2. Where should the flag be positioned in a fair game? Describe how you found the position. Explain why this method works.

3. Will your solution change if the field were in a different shape, like a rectangle, circle, or blob? Explain why or why not.

EXPLORATION 5-7. SUM OF ANGLES IN A POLYGON

Objective: Find the relationship between the number of sides in a polygon and the sum of its angles

1. CREATE A NEW DOCUMENT with the variable polysum.

2. Create a triangle, measure all of its angles, calculate their sum and record it below. Drag around a vertex to confirm the angle sum remains constant.

3. Clear the screen and repeat this procedure for polygons up to six sides and record the information below.

name of polygon number of sides sum of anglestriangle 3quadrilateral 4pentagon 5

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hexagon 6

4. There are two types of formulas which will predict the sum of the measures of the angles in a polygon. A recursive formula uses the constant difference in the sum of angles for every increase in one side of the polygon.Fill in the blank: Sum of angles in polygon = Sum of angles in polygon + ________˚

with n-sides with (n-1) sidesUse the recursive formula to determine the sum of the angles in a heptagon, a seven-sided polygon. Show your work below.

5. An explicit formula predicts the sum of the angles in any polygon, even if you do not know the previous polygon angle sum. Let n represent the number of sides in any polygon. Use the diagram at the right to think about the value (in terms of n) that represents the number of triangles that share vertices within any polygon. Also, use the table you developed in step 3 to help test your explicit formula.

Fill in the blanks: Sum of angles in polygon = ______ * ( )with n-sides number of

trianglesUse the explicit formula to confirm the sum of the angles in a heptagon that you found in step four. Show your work below.

6. Clear the screen, create a heptagon, measure its angles, and check your answers to questions 4 and 5.

EXPLORATION 6-1. REFLECTION-SYMMETRIC FIGURES

Objective: Determine properties of reflection-symmetric figures.

1. CREATE A NEW DOCUMENT with the variable refsym.

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2. Create a vertical line on the screen. Draw the left hand side of a face with polygons, circles, and arcs as shown to the right.

Create an arcPress and choose 2:Arc, move the pencil to a starting position, press , move the pencil a little more, press , and move the pencil one more time until the desired arc, and press . Move the endpoints of the arc if you would like to adjust its length. Move the middle point of the arc if you would like to adjust its curvature.

3. Reflect each of the objects over the line until your face is complete.

4. Adjust the size and shape of the ears, eyes, nose, and lips on the LEFT SIDE of the screen and observe the changes on the right side. For smoother movement, you may want to move the pointer on top of a point, press , to LOCK, and drag using the keypad. Press to return to the pointer when you are finished dragging.5. The vertical line is called a line of symmetry. The face is reflection-symmetric about this line. State at least two properties of a figure that is reflection-symmetric (without using reflection or symmetric in your explanation).

6. Drag the symmetry line until it is oblique and the face looks distorted. Construct a segment with the middle of the eyes as endpoints. What geometric property will the symmetry line have on this segment? Verify this conjecture by making appropriate measurements and/or calculations. Describe your findings below.

EXPLORATION 6-2. ISOSCELES AND EQUILATERAL TRIANGLES

Objective: Determine properties of isosceles and equilateral triangles.

1. CREATE A NEW DOCUMENT with the variable isostri.p. 42


2. Create scalene ∆ABC (with unequal side lengths), measure and label segments AB and AC as shown below to the left.

3. Construct the midpoint D of side BC. Construct the median of BC (the segment connecting A and D), the perpendicular bisector of BC, and the angle bisector of < CAB as shown in the diagram above to the right.4. Drag the vertex A until the median, perpendicular bisector, and angle bisector coincide. What type of triangle results? Explain below how you determined this result.

5. Measure the base angles, < ACB and < ABC. Write an IF...THEN... statement below describing the condition in step four and your results in step five.

6. Create the perpendicular bisectors of the other sides of the triangle, segments AC and AB. Drag the vertices of the triangle until the triangle is reflection symmetric with respect to all of the perpendicular bisectors. 7. Measure the remaining side and angle in the triangle. What type of triangle results? Explain below how you determined this result.

EXPLORATION 6-2P. SHARK ATTACK.

Objective: Determine the ideal position within an equilateral triangle given a series of conditions.

1. CREATE A NEW DOCUMENT with the variable shark.

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Guppy, Tadpole, and Goldfish beach surround Equilateral Sea, a perfect equilateral triangle. Tiger shark and his family swim together in the Equilateral Sea. When the family gets hungry, Tiger makes sure they stay in one place (so he doesn't lose them), and then finds food at each of the beaches. Tiger gets food at one beach at a time because he can only hold so much food in his mouth at a time. So he will get food at one beach, come back to share the food (small square inside the triangle), and then repeat the process at the other beaches.

2. Use rotations to construct Equilateral Sea, and place a point inside the triangle to represent the hungry sharks. Construct perpendicular lines from the sharks to each of the sides of the triangle and the intersection points on the beaches.

3. Where should Tiger place his family in the Equilateral Sea in order to swim the least distance for his three hunting trips? Explain your reasoning.

4. What is the sharks lived in Isosceles Sea? Would your results from step 3 change? Why or why not? Explain your analysis.

EXPLORATION 6-3. CONSTRUCTING PARALLELOGRAMS

Objective: Construct a parallelogram and rectangle. Modify the parallelogram into a rhombus and the rectangle into a square.

1. CREATE A NEW DOCUMENT with the variable pararect.2. Create an oblique line and a parallel line approximately 3 cm away. Connect the points on each line with a segment as shown in the diagram below to the left.

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3. Construct a point on the line to the left. Construct a line parallel to the segment passing through this point, and create a point of intersection as shown in the diagram above to the right.

4. Label the vertices M, A, T, and H. Construct overlapping segments and hide the three lines until your parallelogram matches the one to the right. It's quickest to hold down the shift key, select all three lines, and then choose the hide command. Remember to his after you hide the lines.

5. Measure the slopes of the sides of the figure. Drag the vertices around until the slopes change. Is the figure still a parallelogram? Explain how you know.6. Delete the slope measurements. Measure the length of each side of MATH. Drag the vertices until each of the sides have the same length. This special type of parallelogram is called a rhombus.

Once you drag the vertices again, the parallelogram is no longer a rhombus because the sides will not have the same length. Your goal in chapter seven of the text is to construct a rhombus once you familiarize yourself with its properties.

7. Clear the screen. Create an oblique line and then construct a perpendicular with a point of intersection C. Place point U on one line and point S on the other line as shown below to the left.

8. Draw perpendicular lines passing thru points S and U. Label the point of intersection B as shown in the diagram above to the right.

9. Create overlapping segments CU, UB, BS, and SC. Hide the four lines. Choose Hide/Show, and then select the four lines by pressing ENTER.

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10. Measure the length of each side of the figure CUBS. Drag points U and S until the measurements of all of the sides are equal. The most specific name for this figure is a square because all of its angles and sides are equal. This quadrilateral can be named a parallelogram, a rectangle, and a rhombus. Make necessary measurements on the calculator to justify why these names are also applicable to the figure. Explain below.

11. Once you drag the vertices again, the rectangle is no longer a square because the sides will not have the same length. Clear the screen and use rotations to construct a square that will always remain a square even after you drag around the endpoints. Explain the steps to your construction below.Recall that you must create a numerical edit (under ) for the number of degrees you wish to rotate the object.

EXPLORATION 6-4. CONSTRUCTING A KITE

Objective: Construct a kite using three different methods.1. CREATE A NEW DOCUMENT with the variable kite.2. Construct a kite using two intersecting circles as shown in Figure I of your text on page 323. Hide the circles and leave only the kite. Make necessary measurements on the calculator to justify why this is a kite. Explain below.

3. Clear the screen. Construct a kite using a reflected triangle over a symmetry line as shown in Figure II of your text on page 323. Make necessary measurements on the calculator to justify why this is a kite. Explain below.

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4. Modify the vertices of the kite to make it convex (if necessary). Create the second diagonal of the kite and label the intersection of the diagonals F as shown in the diagram to the right.

5. Explain in a complete sentence below how the diagonals of a kite relate to each other. Make necessary measurements on the calculator to justify your answer below. Differentiate the diagonals by calling CD the symmetrical diagonal.

6. Clear the screen. Construct another kite using a different method based on your explanation in step 5. Use an exploration from chapter five for additional assistance. Explain the steps to your construction below.

EXPLORATION 6-5. CONSTRUCTING A TRAPEZOID

Objective: Construct a trapezoid and an isosceles trapezoid, and explore their properties.

1. CREATE A NEW DOCUMENT with the variable trapez.

2. A trapezoid is a quadrilateral with at least one pair of parallel sides. Construct a trapezoid that is not a parallelogram. Explain the steps to your construction below. Make necessary measurements on the calculator to justify why this is a trapezoid and explain below.

3. Measure and label a pair of consecutive angles between a pair of parallel sides, as shown in the diagram to the right.

4. Drag the endpoints of the original segment until you see the angles change measurement. What relationship exists between these angles?

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5. Clear the screen. An isosceles trapezoid is a trapezoid that has a pair of base angles equal in measure. Construct an isosceles trapezoid using the Isosceles Trapezoid Symmetry Theorem and the drawing related to the theorem on page 330 of your text. Hint: Use a Reflection. Explain the steps to your construction below.

6. Measure the length of the legs, the non-parallel sides, of the isosceles trapezoid. Drag around the vertices until the measurements change. What relationship exists between the legs of an isosceles trapezoid?

EXPLORATION 6-M. MYSTERY QUADRILATERALS

Objective: Determine the most specific name for each mystery quadrilateral (parallelogram, rectangle, square, rhombus, trapezoid, isosceles trapezoid, kite, or quadrilateral).

You must provide a valid “proof” for each mystery quadrilateral. For each quadrilateral, write a short paragraph (see below for an example). Support your answer with the measurements you have taken. Make sure that you list all relevant measurements on your lab. Also be sure to state why the quadrilateral could NOT be any other name.

Before getting started, fill in your hierarchy to refer back to if needed!

Quadrilateral

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Example Solution Paragraph:

Quadrilateral ABCD is a kite because consecutive sides are congruent. Opposite sides are NOT parallel because their slopes are different. Even though there is a right angle, ABCD can not be a rectangle if it is not a parallelogram.

m BC = 3.06 cm

m BA = 3.06 cm

m AD = 2.40 cm

m DC = 2.40 cm

Slope AD = -3.54Slope BC = -12.81Slope DC = 0.28Slope BA = 0.51

m ADC = 90.00°

A

B

C

D

1. Open QUAD1 in the MYSTERY folder. MYSTERY QUADRILATERAL #1

Solution Paragraph. sketch of diagram and relevant measurements!



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EXPLORATION 6-6. ROTATION SYMMETRY

Objective: Compare the number of vertices in an object to its angle of rotation.

1. CREATE A NEW DOCUMENT with the variable rotsym.

2. Place point A in the middle of the screen. Create an arc, starting at point A, and ending near the top of the screen. Create and measure an angle in the lower left-hand side of the screen as shown in the picture to the right.

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3. Rotate the arc about point A using the angle. Repeat the arc rotations until the arc has gone completely around (make seven arcs). Note that the arcs may not perfectly overlap.

4. Create a seven sided polygon (heptagon) which has vertices touching the ends of each arc as shown in the diagram to the right.

5. Adjust the angle in the lower-left hand side of the screen until there is exactly six spokes/vertices as shown above in the picture to the right (two arcs/vertices overlap). The arcs do not have to be perfect, but should be close to overlapping. Record the information on the next page.

#arcs/vertices name of polygon degree of rotation6 hexagon5 pentagon4 quadrilateral3 triangle

6. Adjust the angle in the lower-left hand side of the screen until there are exactly five spokes/vertices. Record the information above. Repeat the procedure until the table above is complete.

7. The overlapping arcs/vertices occur in this exploration because you are creating a special type of polygon. Make up a name that you would use to classify these polygons.

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Make necessary measurements on the calculator to justify why you have chosen this name. Explain below.

8. Write an equation below relating the number of vertices (V) in this special type of polygon compared to the degree rotation (D) between consecutive vertices in the polygon. Explain how you determined this.

EXPLORATION 6-7. REGULAR POLYGONS

Objective: Determine properties of a regular polygon

1. CREATE A NEW DOCUMENT with the variable regpoly.

2. Create a regular pentagon that will cover a large portion of the screen as shown to the right. Press and choose 5: Regular Polygon, move the pencil to the center of the screen, press to locate the center, move out from this center (you'll see a dotted circle) as far as you can before part of the circle goes off the screen. Press . Move the pencil away from the point on the dotted circle until the screen reads {5} and press . You will see a regular pentagon as shown to the right.

3. Measure the side lengths and angles of the regular pentagon. Explain below two properties of regular polygons which you discovered.

4. Connect seven segments from the center of the pentagon to each of its vertices. Five triangles should form as shown in the diagram to the right.Do you know which company uses this design as their logo? 5. Without measuring on your calculator, determine the measure of each of the three angles in one of these triangles. Explain your reasoning and show your work below. Hint: Look back at Exploration 6-6.

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6. Check your answer to step five by measuring each angle of one of the triangles. If your answers do not match, go back and check your work.

EXPLORATION 7-2. CONGRUENT TRIANGLES

Objective: Determine arrangements to determine congruent triangles1. CREATE A NEW DOCUMENT with the variable congtri.2. Create two different looking triangles TRI and AGN as shown to the right.3. Measure the lengths of the three sides in both triangles. Move the vertices of ∆TRI until TIAN, TRAG, and RIGN. It's OK if you are within 0.01 cm away. Is this the only possible arrangement for the triangle to satisfy this condition (excluding rotations)? Two figures are if their corresponding sides AND angles are . You have satisfied the condition that the corresponding sides are ; now you must verify that the corresponding angles are also .4. Measure and compare the corresponding angles of the two triangles. Given that three corresponding sides are congruent (SSS), is this enough to verify that ∆TRI∆AGN? Compare

your results with a classmate.

5. Clear the screen. Recreate triangles TRI and AGN as shown in step two. Measure the angles in ∆TRI until <RTI<NAG, <RIT<NGA, and <TRI<ANG. It's OK if you are within 0.15˚. Is this the only possible arrangement for the triangle to satisfy this condition (excluding rotations)?

6. Measure and compare the corresponding sides of the two triangles. Given that three corresponding angles are congruent (AAA),is this enough to verify that ∆TRI∆ANG? Compare your results with a classmate.

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7. Clear the screen. Recreate triangles TRI and AGN as shown in step two. Measure the angles and sides in ∆TRI until <RTI<NAG, TRAN, and <TRI<ANG. It's OK if you are within 0.01 cm or 0.15 degrees. Is this the only possible arrangement for the triangle to satisfy this condition (excluding rotations)?

8. Measure and compare the remaining corresponding sides and angles of the two triangles. Given that two corresponding angles and an included side are congruent (ASA), is this enough to verify that ∆TRI∆ANG? Compare your results with a classmate.

9. Clear the screen. Recreate triangles TRI and AGN as shown in step two. Measure the sides and angles in ∆TRI until TIAN, <TIRANG, and RIGN. It's OK if you are within 0.01 cm or 0.15 degrees. Is this the only possible arrangement for the triangle to satisfy this condition (excluding rotations)?

10. Measure and compare the remaining corresponding sides and angles of the two triangles. Given that two corresponding sides and an included angle are congruent (SAS), is this enough to verify that ∆TRI∆AGN? Compare your results with a classmate.

11. Clear the screen. Recreate triangles TRI and AGN as shown in step two. Measure the angles and sides in ∆TRI until <RTI<NAG, <TRI<ANG, and RING. It's OK if you are within 0.01 cm or 0.15 degrees. Is this the only possible arrangement for the triangle to satisfy this condition (excluding rotations)?

12. Measure and compare the remaining corresponding sides and angles of the two triangles. Given that two corresponding angles and a non-included side are congruent (AAS), is this enough to verify that ∆TRI∆ANG? Compare your results with a classmate.

EXPLORATION 7-5. SSA CONDITION AND HL CONGRUENCE

Objective: Examine triangles in the SSA condition and verify the condition for HL congruence.1. CREATE A NEW DOCUMENT with the variable ssahl.2. Create a circle with center A, and radii AB and AC, as shown on the next page.

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3. Create a line passing thru points B and C. Create segment AD from A to a point on the line, left of point B, as shown in the diagram to the right of the previous page.4. Create overlapping segment CD. Hide the line and make the circle dotted as shown in the diagram to the right.

5. ∆ABD and ∆ACD are overlapping. Mentally separate and redraw the triangles below so they are not overlapping.

6. Place tick marks on the triangles to identify pairs of congruent angles and congruent sides (you should have three). Justify below why each pair of corresponding parts is congruent.

7. Given that two corresponding sides and a non-included angle are congruent (SSA), is this enough to verify that ∆DBA∆DCA? Explain.

8. ∆DBAand ∆DCA will be congruent under one situation. Drag point B around the circle until the two triangles are congruent. When ∆DBA∆DCA, what type of triangles are on the screen? Explain your reasoning below.

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9. Re-evaluate your explanation in step 7. Explain below why this might be called HL congruence instead.

10. Wait! There's even another case. Construct a Star Trek symbol inside a circle as shown to the right. Modify point S until <AST=<ASR (or within 0.15˚).

11. Place tick marks on the triangles to identify pairs of congruent angles and congruent sides without any additional measuring (you should have three). Justify below why each pair of corresponding parts is congruent.

12. Measure and compare the remaining corresponding sides and angles of the two triangles. Given that two corresponding sides and a non-included angle are congruent (SsA), is this enough to verify that ∆DBA∆DCA? Explain.

This case in step 12 worked and the case in step seven did not work because the sides opposite the congruent angles in each triangle were longer than the other pair of congruent sides. Therefore, SsA will be used to distinguish step 12 from step 7 (SSA).EXPLORATION 7-6. TESSELLATIONS

Objective: Construct tessellations with triangles using rotations and translations.

1. CREATE A NEW DOCUMENT with the variable tessell.

2. Follow along with Activity 1 on page 398 of your text. Create ∆ABC near the lower left side of your screen. Create the midpoint M on side AC as shown below to the left.

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3. Create a Numerical Edit for rotation purposesPress and choose 6:Numerical Edit. Move the pencil near the upper-right hand portion of the screen, press , type 180, press . This will be used for rotating the ∆ABC by 180˚.

4. Rotate ∆ABC about point M by 180˚, as shown above in the diagram to the right.

5. Create overlapping vector BC with initial point B and terminal point C. Press and choose 7: Vector. Select point B first, then point C. This will allow you to move the figure to the right.

6. Translate ∆ABC by vector BC. Press and choose 1: Translation. Select the triangle and then the vector. Then translate the rotated triangle by vector BC. Continue the process until you have created eight triangles total, as shown to the right. Drag around the vertices of the triangle to see if the triangles remain connected.

7. Create overlapping vector BA with initial point B and terminal point A. This will allow you to move the figure up.8. Translate ∆ABC by vector AB. Translate the rotated triangle by vector AB. Continue the process until you have created two rows with sixteen triangles total, as shown to the right. Drag around the vertices of the triangle to see if the triangles remain connected.

9. Which kind of special quadrilateral is ABCD? How do you know?

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10. Clear the screen. Try to build the tessellation to the right. Use Activity 2 on page 399 of your text as a guideline for hints. Drag around the vertices of ∆XYZ to see if the tessellation remains connected.

EXPLORATION 7-7. PROPERTIES OF PARALLELOGRAMS

Objective: Discover properties of parallelograms.

1. CREATE A NEW DOCUMENT with the variable pgram.

2. Construct parallelogram CUBS. Make sure the figure remains a parallelogram after you drag around its endpoints. Don't worry about doing overlapping segments and hiding the lines. Just create intersection points. Describe how you constructed your parallelogram.

3. Measure the lengths of the two pairs of opposite sides. Drag around one of the vertices until you see the measurements change. Write a property of parallelograms below based on your observations.

4. Delete the length measurements. Measure the two pairs of opposite angles. Drag around one of the vertices until you see the measurements change. Write another property of parallelograms below based on your observations.

5. Delete the angle measurements. Create the diagonals of the parallelogram. Make necessary measurements on the calculator to determine a property of the parallelogram's diagonals. Hint: Create a point at intersection between the two diagonals and investigate the lengths of each part of the diagonals. Drag around one of the vertices to make sure the property is true in general.

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EXPLORATION 7-7P. BOUNCING OFF THE WALLS

Objective: Determine the relationship between parallel lines and a position on the triangle.

1. CREATE A NEW DOCUMENT with the variable bounce.

Jeff is in the triangular room shown to the right. He walks from a point on AC parallel to BC. When he reaches AB, he turns and walks parallel to AC. When he reaches BC, he turns and walks parallel to AB. He continues this process until he returns to his starting point.

2. Construct a diagram to represent this situation.Make sure the lines remain parallel to the sides, even after you drag around the vertices of the triangle or change Jeff's starting point. Create segments to overlap the parallel lines and hide the lines that fall outside of the triangle.

3. What is the smallest number of walls that Jeff will touch before he reaches his starting point? What is his starting point in this situation? Why will this specific starting point result in the least number of bounces?

4. What is the largest number of walls that Jeff will touch before he reaches his starting point? Will Jeff always wind up in his original starting position? Explain why or why not.

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EXPLORATION 7-9. EXTERIOR ANGLES

Objective: Compare angles and sides in a triangle, and explore the relationship between interior and exterior angles.

1. CREATE A NEW DOCUMENT with the variable extang.

2. Construct ∆CAT on top of an oblique line with exterior point S, as shown in the diagram to the right. Measure the sides and angles of ∆CAT.

3. Fill in the table below comparing each side of the triangle with its opposite angle. Use YOUR measurements.

Side Length Opposite Angle Measurement

AT= ACT= .

AC= ATC= .

CT= CAT= .

Rank the three side lengths 1,2, and 3 from longest to shortest. Next, rank the angle measurements opposite those sides 1,2, and 3 from largest to smallest. What do you observe?

Now test this hypothesis by dragging around a vertex and seeing if it remains true. Does it?

4. Measure exterior angle ATS. Describe below how this angle compares with the measurements of alternate interior TAC and TCA. Drag around point T to see if this is still true.

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5. Compare the exterior angle with the sum of its alternate interior angles.

Exterior Angle Measurement Sum of Interior Angles at other two vertices

ATS = ACT + CAT =

Drag around the vertices of the triangle and try it again. Is the outcome the same? Explain below why or why not this outcome would be the same. Hint: Compare the sum of the angles in a triangle to a straight angle.

EXPLORATION 8-1. PERIMETER OF A REGULAR POLYGON

Objective: Determine the perimeter of a regular polygon with n sides.1. CREATE A NEW DOCUMENT with the variable regperim.2. Use the regular polygon tool to create an equilateral triangle. Measure the perimeter and length of one side (distance between consecutive vertices), and record the information in the table below.Name of regular polygon number of sides length of one side perimeter

equilateral triangle 3 __________ __________

square 4 __________ __________

hexagon __________ __________

octagon __________ __________3. Repeat the procedure in step 2 with a square, regular hexagon, and regular octagon. Record the information in the table above.4. Is there a pattern in the values in the table above? Explain below.

5. If the length of one side of a regular n-gon is represented by s, state its perimeter p in terms of n and s in the form of an equation.

EXPLORATION 8-2. AREA OF A RECTANGLE

Objective: Determine the area of a rectangle given a length and width.

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1. CREATE A NEW DOCUMENT with the variable rectarea.2. Construct a rectangle without hiding any of the lines. Use the polygon tool to create an overlapping rectangle so the vertices of the polygon meet the vertices of the rectangle that you are creating. The polygon tool allows you to measure area. 3. Hide the lines which created the rectangle. Only the polygon should be on the screen. Dragaround the vertices of the polygon to ensure the figure remains a rectangle.

4. Measure the lengths of two consecutive sides, and the area, of the polygon as shown in the figure at the right.

5. Use either addition, subtraction, multiplication, or division, with the calculate tool to determine a relationship between the length, width, and area of the rectangle. Write this relationship below in a formula.

EXPLORATION 8-2P. OPTIMAL QUADRILATERALS

Objective: Determine the quadrilateral with fixed perimeter that has maximum area.

1. CREATE A NEW DOCUMENT with the variable optimal.

2. Create a quadrilateral using the polygon tool.

3. Measure the perimeter of the quadrilateral and drag the vertices until the perimeter is 8.0 cm.

4. Measure the area of the quadrilateral, and record the value in the table below. Drag the vertices of the quadrilateral until it changes shape, but still has a perimeter of 8.0 cm. Try various types of quadrilaterals and record the new area below. Continue this process for a variety of different quadrilaterals.

Quadrilateral Area (perimeter = 8 cm)

Quadrilateral __________

Rectangle __________

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Parallelogram __________

Square __________

Kite __________

____________ __________

5. What type of quadrilateral with perimeter 8.0 cm will have a maximum area? Justify your response.

6. Repeat the exploration using a different fixed perimeter, how do your results change? Explain your analysis.

7. CREATE A NEW DOCUMENT with the variable optimal2.

8. Suppose the exploration is reversed so that the area of a quadrilateral is fixed to a specific number, like 8.0 cm2. What type of quadrilateral with fixed area will have a minimum perimeter? Explain your constructions and your analysis.

EXPLORATION 8-3. AREAS OF IRREGULAR POLYGONS

Objective: Determine the area of an irregular polygon on a coordinate plane.

1. CREATE A NEW DOCUMENT with the variable irrarea.

2. Use the Format tool to turn on the coordinate axes to a rectangular grid. Change the unit label on the axes to equal 1 as shown below to the left.

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3. Create a large W on the screen using the coordinates shown above to the right. Make sure that each vertex falls on a grid point.

4. Determine the area of each of the pieces by exact calculation or estimation. Keep in mind that the horizontal and vertical distance between two consecutive vertices is one unit. Do not redraw these figures on your calculator - just examine the figures to the right.Area of A=Area of B=Area of C=Area of D=

5. Now use this method to find the area of the letter W. For instance, the region identified in the picture to the right might be about 0.7 cm2. Use small pieces and the coordinates on the diagram to the right to estimate the area of the W. Show your work below.

6. Change the unit label on the axes to equal 0.5 (otherwise, the area will not be measured accurately). Use the Area tool to find the exact area of the W. Check your results with your work in step 5. If your answer is not within 0.7 cm2, then go back and check your work from step 5.

A more precise approach to finding the area of the W will be explored in section 8-5.

EXPLORATION 8-4. AREA OF A TRIANGLE

Objective: Determine the area of a triangle and relationships between the altitudes of a triangle.

1. CREATE A NEW DOCUMENT with the variable triarea.

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2. Use the Format tool to change the coordinate axes to a rectangular grid and turn the grid ON. Change the unit label on the axes to 1. Create ∆ABC with vertices shown to the right.

3. Use the estimation methods in section 8-3 to determine the area of this triangle. Mark up the diagram to the right and show your work below.

4. Change the unit label on the axes to equal 0.5. You may need to move the origin a little lower to see the entire triangle. Use the Area tool to check your answer. Check your results with your work in step 3.

5. The triangle has three bases and three heights (altitudes). You can find the area of a triangle with any one pair of base and height. Choose one side of the triangle, and call it the base (AB=3 cm). The height of the triangle with base AB will be the perpendicular distance from AB to the opposite vertex C (height = 2 cm).

If the position of point C changes along the y-axis, the height (altitude) of the triangle always remains the same, as shown in the diagrams to the right.

Use the formula of a rectangle and the picture to the lower right to determine the formula for the area of a triangle. Write the formula below and explain how you derived it.

You will check this formula by creating a different altitude and measuring its corresponding base.

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5. Create an altitude from B to ACCreate a line perpendicular to AC and passing through B. Call the point of intersection D. Create an overlapping segment BD, and hide the line.

6. Measure and label the lengths of base AC and height BD.

7. Use the Calculate tool to check your area formula in step four with base AC and altitude BD.

8. Construct the remaining altitudes EC and AF as shown to the right. Hide the perpendicular lines that you used to construct the altitudes.

9. Drag the vertices of ∆ABC around and describe below the relationship among the altitudes.

10. The altitudes intersect at a point called the orthocenter. The orthocenter lies inside an acute triangle, as shown in the example above. Sometimes, the altitudes lie on, or outside, the triangle. In the picture at the right, altitude BD is inside the triangle, but the other two altitudes and the orthocenter lie outside of the triangle.

You must unhide, or show, the perpendicular lines in order to see an altitude outside of the triangle. Press and choose 1:Hide/Show, move the pointer on top of each dotted perpendicular line and press when the screen reads THIS LINE. You should now have three perpendicular lines showing.

11. Drag the vertices of ∆ABC around and observe changes in the location of the altitudes and orthocenter. Name the type of triangle that has two altitudes outside of the triangle and the orthocenter lying outside of the triangle.

Name the type of triangle that has two altitudes on the triangle and the orthocenter lying on of the triangle.

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EXPLORATION 8-4P. TRIANGLE IN A RECTANGLE

Objective: Determine the ratio of the areas between a special rectangle and triangle.

1. CREATE A NEW DOCUMENT with the variable trirec.

2. Construct rectangle ABCD with length to width in the ratio of 3:2. As you drag the vertices, the ratio should always remain 3:2. (Hint: use reflections). Explain how you created this construction.

3. Create angle bisectors AN and BM of angle A and angle B, respectively. Make E the point of intersection of the angle bisectors. You should end up with a small triangle EMN at the bottom of the rectangle, as shown to the right.4. What is the ratio of the area of the area of rectangle ABCD to the area of triangle EMN? Is this always true, even when you change the size of the rectangle? Explain your analysis.

5. Let segment length MN equal 2x. Divide the rectangle in smaller sections and prove your results from step 4. Show your work.

6. How would the results from this investigation differ if ABCD were a parallelogram? Explain your analysis.

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EXPLORATION 8-5. AREA OF TRAPEZOIDS

Objective: Determine the area of a trapezoid, parallelogram, and an irregular polygon.

1. CREATE A NEW DOCUMENT with the variable traparea.

2. Construct a trapezoid (start with segment OI and construct a parallel line ZD) and with altitudes passing thru D and O, as shown in the diagram to the right.

3. Create segment DO. Make overlapping segments for each of the altitudes and for segment ZD and hide any lines which were used in the construction of the trapezoid. Drag around the vertices to ensure the trapezoid remains intact.

Use and 8: Thick for the four segments which make up the sides of the trapezoid.

4. The altitudes passing thru O and D should be equal in length. Explain why below without measuring.

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5. Use the Comment feature so the altitudes equal h, and the bases are b1 and b2, as shown on the diagram.

The area of trapezoid ZOID is the same as the sum of which two triangles?

Predict the area of the trapezoid below in terms of h, b1, and b2, by using the formula for area of a triangle and your answer to the previous question.

AREA OF TRAPEZOID =

6. Use the polygon tool (and 4: Polygon) to create a quadrilateral which overlaps and contains the same vertices of trapezoid ZOID.

7. Measure the area of the polygon using the Area tool.

8. Find the lengths of b1, b2, and h so that we can check the area given by your trapezoid area formula in step five with the area value on the screen. Use the Calculate tool to determine the area using your formula in #5. Drag the vertices around and observe if the formula continues to match the area measurement. If the numbers do not match, check your trapezoid formula with a classmate.

9. Without making any additional measurements, drag point Z and explain how you know when figure ZOID has been changed into a parallelogram. You may need to slide the entire figure to the right, or make segment OI a little shorter so it fits on the screen.

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10. Once you have turned ZOID into a parallelogram, use the formula for the area of a trapezoid and substitution to express the area of the parallelogram strictly in terms of b (only one variable is needed because b1 = b2 now) , and h. Show how you derived the formula below.

11. In Exploration 8-3, you estimated the area of a large W on the coordinate plane. Now you can find an exact value of the area using a process called encasement.

This means a large rectangle will surround the W. The inside of the rectangle will include a W, two trapezoids, and five triangles. Since you have been introduced to finding the area of a rectangle, triangle, and trapezoid, you should be able to find the area of the irregular polygon W.

Find the area of the W using encasement. The origin of the hidden coordinate axes is placed on the diagram. Show your work below.

EXPLORATION 8-6. PYTHAGOREAN THEOREM

Objective: Verify the Pythagorean Theorem.

1. CREATE A NEW DOCUMENT with the variable pythag.

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2. Construct a right triangle using perpendicular lines. Create overlapping segments and hide the lines.

The length of the smallest side will be called a; The length of the middle-size side will be called b;and the length of the largest side will be called c.

3. Measure the lengths of a, b, and c.

4. Use the Calculate tool to find a2 + b2. You will type in a^2+b^2 because the exponent is represented by a caret, . Also find c2.

5. Drag around the vertices of the right triangle and observe changes in your calculations.

6. State the formula for the Pythagorean Theorem below based on your conclusions from step 5.

7. A rational right triangle is a triangle where all the sides are rational numbers, and one of the angles is a right angle. Find a rational right triangle such that the length of the hypotenuse is numerically equal to the area of the triangle. Explain your analysis.

8. Find a rational right triangle such that the length of the hypotenuse is numerically equal to the area of the triangle and the perimeter of the triangle is a prime integer. Explain your analysis.

EXPLORATION 8-7. CIRCUMFERENCE AND ARC LENGTH

Objective: Determine how the circumference and arc length relate to the radius of a circle.

1. CREATE A NEW DOCUMENT with the variable circumf.

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2. Create a large circle with a center near the center of the screen. Measure the circumference of the circle.

3. Use the Regular Polygon tool and create an equilateral triangle, square, pentagon, and hexagon on the circle with the circle's center. Measure and label the perimeters of each of these polygons and record their values, along with the circle's circumference, using YOUR values in the table below.Regular Figure Number of Sides Perimeter/CircumferenceTriangle _____ __________Square _____ __________Pentagon _____ __________Hexagon _____ __________Circle ??? __________

4. How many sides should an inscribed regular polygon have if its perimeter is the same as the circle's circumference? Justify your answer below by explaining how the number of sides in an inscribed regular polygon relates to the circumference of a circle.

5. Clear the screen. Create a new circle and radius in the middle of the screen. Measure the circle's circumference and radius.

6. Use the Calculate tool and compute the circumference divided by two times the circle's radius (or diameter) [I.e., circumference/(2*radius)]. Make sure that you include parantheses around the 2*r.

Modify the size of the circle and observe changes in your measurements and the status of the calculated value. What value reoccurs in your calculation?

7. Use your observations in step six to write an equation below relating the circumference of a circle, c, compared to its radius, r.

8. Create another radius of the circle and an arc between the two points on the circle. Refer to Exploration 6-1 to create an arc.

Measure the central angle of the sector which contains the arc. Measure and label the length of the arc using the Distance & Length tool (press when the screen reads WHICH OBJECT? and choose "THIS ARC").

9. Change the size of the circle until it has a circumference very close to 10 cm.

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10. Make the arc longer by moving the arc's endpoints further apart. Notice the change in the measurement. Continue changing the arc until its length is equal to one-tenth the circumference of the circle, about 1 cm. Record the measure of the central angle in the table below.

Change the arc length until the length of the arc is equal to one-quarter, one-half, and three-fourths of the circle's circumference. Record the corresponding arc lengths and central angles in the table.

Fraction of Circle Arc Length Central Angle Circumference1/10 __1 cm__ _________ 10 cm1/4 _________ _________ 10 cm1/2 _________ _________ 10 cm3/4 _________ _________ 10 cm

11. Use the table to help you find a pattern (formula) relating the arc length, central angle, and circumference of the circle. Hint: Think of the portion of a circle the arc fills as a fraction of the full circle. Reminder: A full circle measures 360˚ around.

12. Write a formula below for the arc length, L, in terms of the central angle, a, and the circumference of the circle, C and any relevant numbers (like 360). Explain how you derived the formula.

13. Check your formula using the numbers from a line in the table. Show below that your formula works or does not work.

EXPLORATION 8-8. AREA OF A CIRCLE THROUGH DATA ANALYSIS

Objective: Collect data and determine the area of a circle in terms of the radius as a function.

1. CREATE A NEW DOCUMENT with the variable areacirc.

2. Create a medium-sized circle near the center of the screen. Measure and label the radius and area of the circle as shown to the right.

3. Now you are ready to set the calculator up to store data. Press and choose “7: Collect Data” and “2: Define Entry.” Move the cursor over the radius measurement first until it reads THIS NUMBER, and press . The calculator now recognizes that all

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radius measurements will be stored in the first column of the data table.

Move the cursor over the area measurement until it reads THIS NUMBER and press . The calculator now recognizes that all area measurements will be stored in the second column of the data table.

Press and to collect and store a data point. Notice that the lower-left hand portion of the screen will read DATA PLACED IN VARIABLE SYSDATA.

4. Access the stored data by pressing APPS and selecting 6:Data/Matrix Editor » 2:Open (shown below to the left). In the variable option, use the arrow key to choose “sysdata” and press twice (shown below to the right). You should now see the radius value stored in c1 and the area value stored in c2. Note that the calculator automatically labels the columns for you.

Next, you will collect many data points by changing the radius of the circle and then storing each new set of data in “sysdata” by pressing and .

5. You can view the transfer of information from Geometry to the data editor by setting up a split screen. Press and then . Use the arrows and key to set the Split Screen to “LEFT-RIGHT,” with the Split 1 App as Geometry and Split 2 App as Data/Matrix Editor. Press and the Split Screen should appear. You may need to press and for both applications to appear on the screen. The and sequence will also toggle you back and forth between the geometry program and the data editor. You will need to stay on the geometry side to continue collecting data.

6. On the geometry side, adjust the circle so the radius and area measurements change. Press and to collect and store another data point. Notice that the data is immediately stored in the “sysdata” editor on the right side of the screen. Repeat this process with about 10 more circles. Make sure you include very small and very large circles in your data collection.

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7. View the data on a graph by first setting up the STAT PLOT menu in the Y= editor. Press and select 2:Y= Editor. Move the cursor up to “Plot 1:,” and press . Use the arrow key to set the x-value as data in column c1 and the y-value as data in column c2 and press twice. View the data on a graph by pressing and selecting 9: ZoomData.

8. Move back to the data table by pressing and. Use the menu to find a best fitting model. Choose a “calculation type,” set the x-value as data in column c1 and the y-value as data in column c2, and store the regression equation (RegEQ) to y1(x). Press and the regression function should appear on the screen.

9. Write an equation of the regression function below using three decimal place accuracy based on the values of the regression equation.

10. The leading coefficient in your equation is an approximation of a special number. Write an equation relating the area of a circle (A) compared with its radius (r) using this special number (and not the approximation).

11. Press and to view the graph of a model for the data. Explain in a sentence below how the data relate to the curve.

EXPLORATION 8-8P. AREA OF TANGENT CIRCLES

Objective: Determine the constructions and solutions to problems related to the area of a circle.

1. CREATE A NEW DOCUMENT with the variable tancirc.

A circle is inscribed in a semicircle. That meansthe smaller circle passes only through thesemicircle's center and point on its circumference.

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2. Describe the steps to construct this diagram so that you can change its size, but still keep the smaller circle inscribed in the semicircle.

3. What is the ratio of the area of the semicircle to the inscribed circle? Is this always true? After computing the ratio on your screen, try to prove this conjecture by letting the radius of the inscribed circle equal r. Show your work below.

4. CREATE A NEW DOCUMENT with the variable tancirc2.

Two larger circles equal in radii pass througheach others centers. A smaller circle can be created inside the overlapping region so that itis tangent to the other circles. (Tangent meansthat the circles touch each other, but do notcross over each other, nor do they leave anygaps.)

5. Describe the steps to construct this diagram so that you can change its size, but still keep the smaller circle tangent to the larger circles.

6. What is the ratio of the larger circle to the smaller circle? Is this always true? After computing the ratio on your screen, try to prove this conjecture by letting the radius of the smaller circle equal r. Show your work below.

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7. CREATE A NEW DOCUMENT with the variable tancirc3.

Along the diameter of a circle, you can construct circles equal in radius length that are tangent to each other and tangent to the circle. That is, the circles touch each other, but do not cross over each other, nor do they leave any gaps.

8. Describe the steps to construct an even number of tangent circles so that you can change its size, but still keep all of the circles tangent. How would your construction change if there were an odd number of tangent circles?

9. How does the combined area of all of the n smaller circles relate to the the area of the entire circle? Give your answer in terms of n. Explain your reasoning.

10. After computing the ratio on your screen, try to prove this conjecture by letting the radius of the smaller circles equal r. Show your work below.

EXPLORATION 9-5. FAMOUS PATHS IN GEOMETRY

Objective: Create three different geometric "designs" and relate them to materials studied in the context of plane sections.

1. CREATE A NEW DOCUMENT with the variable path1

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2. Draw a line segment AB. Draw another segment with endpoint D on segment AB and endpoint C above line segment AB.

3. Construct a perpendicular to segment AB passing thru point D. Construct the perpendicular bisector to segment CD. Mark the Intersection Point, P, as shown to the left.

4. Drag point D back and forth along line segment AB. What do you notice is the path of point P? Take a guess if you're uncertain.

5. A good way to check the validity of your guess is to have the calculator Trace the path of point P for you while point D moves back and forth. Before we do this, though, let's hide the two lines through P and segment CD to obtain a clearer picture. Now press and select 2: Trace On / Off and select point P. Finally, select animation by pressing and choosing 3: Animation. Next, click on the hand tool, and drag D toward B. You will see a spring form. (The longer the spring when you release the hand tool, the faster point D will be animated along the segment.) Hit escape to stop. What kind of curve (or portion of curve) seems to have been generated?

6. CREATE A NEW DOCUMENT with the variable path2.

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7. Draw a circle with center C. Place a point B inside the circle. Place point A on the circle (use Point on Object).

8. Construct segment AB. Construct the perpen-dicular bisector of segment AB. Draw a line through points A and C. Name the intersection point of line AC and the perpendicular bisector of AB point P. See the diagram below.

9. Drag point A along the circle. What do you think will be the path of point P? Guess if you're not sure.

10. To test your guess, we will have the calculat-or Trace the path of point P for you while point A moves along the circle. Before we do this let's hide some non-essential clutter as we did earlier. Hide the two lines and the segment so your screen matches the picture to the right.

Select point P to be traced. Animate point A. Recall that you do not hit enter to select point A -- you use the hand tool and drag it to one side (you'll see the spring). Hit escape to stop.

What kind of curve seems to have been generated?

11. Choose Undo. Your curve should be gone. For the third and final path, simply drag point B to a point outside the circle.

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12. When you move point A along the circle, what do you think seems to be the path of point P?As before, guess if you're not sure.

13. Well, let's get a better look. Trace point P and animate A once again. What kind of curve seems to have been generated?

EXPLORATION 10-6. CREATING A TOOLBAR TO REFERENCE FORMULAS

Objective: Write a program that create a toolbar to reference lateral area, surface area, and volume formulas.1. Enter a New Program Press and choose 7:Program Editor » 3:New. Save the file in the main directory folder your name as the variable, and press twice as shown below.

2. You are now in the programming entry screen when your screen matches the diagram to the right (replace your name with george).

The program that you are about to create will set a custom title bar where you can access geometry formulas.

3. Press and choose 7:Custom...EndCustm. The program contains a block of statements between "Custom" and "EndCustm." Each of those statements is either a "Title" (of a pull-down menu) or "Item" (entry in a pull-down menu) statement.

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4. Create each Title by pressing and choose 1:Dialog » 7:Title (or type the word Title), and then type "Prism". Press to move to the next line.

Repeat this procedure for Cylinder, Pyramid, Cone, and Sphere, as shown to the right.

5. Check the status of your program by pressing to return to the home screen. Type the name of the program with parenthesis, yourname(), press until the screen reads "Done," (as shown below to the left) and press to activate the custom toolbar (as shown below to the right). Notice how the toolbar changes at the top of the screen. Press to return to the original title bar.

6. The next step will include the items in the pull down menu. Press and choose 7:Program Editor »1:Current to return to your program. Move the cursor to the right of "Prism" and press to create a new line as shown below to the left.

7. Create a listing by pressing and choose 1:Dialog » 8:Item (or type the word Item). Type "LA = ph." Press to advance to the next line and make the lists SA = LA + 2B and V=Bh, as shown above to the right.

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8. Go back to the main screen, re-enter the program name, and look at the custom menu. Press to view the prism formulas you have just entered.

9. Your goal is to finish the toolbar until there are formulas under each menu. Your formulas should be as specific as possible. For example, the volume of a cylinder should not be V=Bh, but instead V=πr^2h, because you can substitute πr^2 for B, the area of the base.

Go back to the program and start with the next Item under the Cylinder Title. When you have completed the program, try it out. There should be a total of 14 formulas.

p. 83


EXPLORATION 11-6. DISTANCE FORMULA

Objective: Verify and prove the distance formula.

1. CREATE A NEW DOCUMENT with the variable distform.

2. Use the Format tool to turn on the coordinate axes to a rectangular grid.

3. Plot a point in quadrant II and another point in quadrant IV. Measure and label the distance between the points.

4. Measure the coordinates of the two pointsPress and choose 5:Equation & Coordinates, move the pencil on top of one of the points until the screen reads COORDINATES OF THIS POINT, press and repeat the process with the second point.

5. Use the Calculate tool to verify the measured distance using the distance formula:d =

The calculation strip should compare with the diagram to the right. The locations of the values of a, b, c, and d on the screen, respectively correspond with x2, x1, y2, and y1 in the formula.

Press to return to the pointer. Drag the points around the screen. If the distance measurement continues to equal the calculated value, then you have verified the distance formula!

6. Connect the two points with a segment. Use horizontal and vertical segments to draw a picture of a right triangle shown to the right. Find the coordinates at the right angle.

Do not move any of the vertices of the triangle because then it will no longer be a right triangle.

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7. Use the coordinates to calculate the lengths of the legs of the right triangle WITHOUT measuring Distance & Length. You may use the calculator for addition and subtraction purposes only. Show your work below.

8. Use these calculations and the hypotenuse to verify the Pythagorean Theorem. Show your work below.

9. Suppose the triangle had coordinates (x2,y1), (x1,y1), and (x2,y2), as shown to the right. Explain below how the Pythagorean Theorem can be used to prove the distance formula:

d =

EXPLORATION 11-7. EQUATIONS OF CIRCLES

Objective: Determine the relationship between the equation of a circle to its center and radius.

1. CREATE A NEW DOCUMENT with the variable eqcirc.

p. 85


2. Use the Format tool to turn the coordinate axes to rectangular and the grid to on. Change the axes unit from 0.5 to 2. Recall calc. must say THIS UNIT and then drag it closer to the origin.

3. Create a circle with center at the origin and a radius of two as shown to the right. Make sure the radius point lands on the grid.

3. Find the equation of the circlePress and choose 5:Equation & Coordinates, move the pencil on top of the circle until the screen reads EQUATION OF THIS CIRCLE, and press .

*The equation should read x2 + y2 = 22, which is really x2 + y2 = 4.

4. Drag the radius of the circle until it equals 0, 4, and 6. Record the equations in the table below.

Center Radius Equation on Calculator Simplified Form(0,0) 0 __________________ __________________(0,0) 2 x2 + y2 = 22 x2 + y2 = 4(0,0) 4 __________________ __________________(0,0) 6 __________________ __________________

5. Use the information in the table to describe below how the radius affects the equation of a circle centered at the origin.

6. Delete the circle. Now you will determine how the location of the center affects the equation of a circle.

Create a new circle with radius equal to 2 and center at coordinates (4,2), as shown to the right. Make sure the radius point lands on the grid.

7. Find the equation of the circle. You may need to drag the measurement back on the screen. The equation should read (x-4)2 + (y-2)2 = 22, which is really (x-4)2 + (y-2)2 = 4.

8. Drag the center of the circle until it has coordinates (-4,4), (-6,-2), and (10,-4). Record the equations in the table below.

Center Radius Equation on Calculator Simplified Form

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(4,2) 2 (x-4)2 + (y-2)2 = 22 (x-4)2 + (y-2)2 = 4(-4,4) 2 __________________ __________________(-6,-2) 2 __________________ __________________

9. Use the information in the table to describe below how the center affects the equation of a circle centered at the origin.

10. State the standard equation of a circle with center (h, k) and radius r. Your equation should not include numbers and should include the variables x, y, h, k, and r.

11. Demonstrate algebraically that your answer in #10 is indeed correct. Hint: Use the distance formula and the fact that the radius is the distance from the center to a generic point on the circle (x,y).

12. Use the standard equation of a circle to complete the missing information in the table below. Use the calculator to check your answers. You will need to change the axes unit to check some of your answers.

Center Radius Simplified FormA. (8,4) 6 ____________________B. (-2,0) _____ (x+2)2 + y2 = 16C. _____ 10 (x-10)2 + (y+5)2 = 100D. _____ _____ x2 + (y-50)2 = 2500

EXPLORATION 11-8. MEANS AND MIDPOINTS

Objective: Determine the relationship between the equation of a circle to its center and radius.1. CREATE A NEW DOCUMENT with the variable mnmid.

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2. Create ∆CUB. Let H be the midpoint of CU and I be the midpoint of CB. Create segment HI as shown to the right. HI, the segment connecting the midpoint of two sides of a triangle, will be referred as the midline to the base UB.

3. Measure the slopes of midline HI and base UB.To measure the slope of UB you will need to create an overlapping segment. The calc. cannot determine slopes of sides of polygons. It can only find slopes for segments.4. Drag around the vertices and observe the changes in measurement. Use a complete sentence to state below the relationship between a midline and its corresponding base without using the word slope.

5. Measure the lengths of midline HI and base UB. Drag around the vertices and observe the changes in measurement. Use a complete sentence to state below the relationship between the length of a midline and its corresponding base.

6. Clear the screen. Use the Format tool to turn on the coordinate axes to a rectangular grid.

7. Create segment CT with endpoints at (3, 1) and (1,0). Create the midpoint M of segment CT. Measure, label, and organize the coordinates of the points, as shown to the right.

8. Drag around point T around the screen and observe the relationship between the midpoint M and the endpoints of segment CM.9. Suppose the coordinates of C was (x1,y1), and T was (x2,y2). What would be the coordinates of the midpoint M? Express your answer in terms of x1, x2, y1, and y2. State your answer and explain your reasoning below.

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10. Use your formula to complete the missing information in the table below. Use the calculator to check your answers. endpoint C midpoint M endpoint TA. (1,1) ________ (3,0)B. (-1.5,0.5) ________ (1.5,-1.5)C. ________ (1.75,-1) (1.5, 0)D. (-3,-1) (-4, -1.5) ________

11. Clear the screen. Construct a triangle andits medians as shown to the right. Theintersection of the medians of a triangle iscalled the centroid. How do the coordinates ofthe centroid relate to the coordinates of thevertices of the triangle? Write an equation todescribe this relationship. Explain how youdetermined this result..

EXPLORATION 11-8P. MIDPOINTS AND AREAS

Objective: Determine the relationship between the area of a triangle and rectangle related to midpoints.

1. CREATE A NEW DOCUMENT with the variable midarea.

2. Construct a triangle connecting the threemidpoints of the sides, as shown to the right.Triangle DEG is called the medial triangle.Use the polygon tool to create a triangle thatoverlaps triangle DEF.

3. What is the ratio of the areas of the larger triangle ABC compared to the medial triangle DEF? Is this property always true? Explain why or why not. Prove your case.

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4. What is the ratio of the perimeters of the larger triangle ABC compared to the medial triangle DEF? Is this property always true? Explain why or why not. Prove your case.

5. How will you results to steps three and four change if you examined the medial polygons inside different polygons with n sides? Explain your analysis.

6. CREATE A NEW DOCUMENT with the variable midarea2.

7. Construct triangle GOR. Constructmidpoints D and E of segments OR and GO,respectively. Construct rectangle MATH (makesure that you hide any lines used in theconstruction). Use the polygon tool to create apolygon on top of GOR and a polygon on topof MATH.

8. Measure the area of triangle GOR and thearea of rectangle MATH. How do they relate?Why is this true? (Hint: think of the midlineED). Explain your reasoning.

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EXPLORATION 12-1. THE TRANSFORMATION SK

Objective: Determine the effects of a segment under a transformation Sk.

1. CREATE A NEW DOCUMENT with the variable skthm.

2. Clear the screen. Use the Format tool to turn on the coordinate axes with rectangular grid.

3. Create segment MN with M near (1.00,0.25) and N at (-0.50,0.50). Measure, label, and move the coordinates until they correspond with the diagram to the right.

4. The next step will be to create a transformation of segment MN under Sk, where k is the magnitude of the transformation.

Create a horizontal segment with length 2 cm in the upper left hand portion of the screen, beginning at a grid point, as shown below in the diagram to the right. Measure the segment and label it with k=. This segment and measurement will represent the magnitude of transformation, k.

5. Create a dilation of segment MNPress and choose 3:Dilation, move the pencil on top of segment MN until the screen reads DILATE THIS SEGMENT, press , move the pencil to the origin until the screen reads WITH RESPECT TO THIS POINT, press , move the pencil to the measurement 2.00 cm (point at the #, not the segment) until the screen reads USING THIS FACTOR, and press .

6. Press to return to the pointer, and don't bother labeling the endpoints of the image M' and N' as shown above to the right. That's just done for your reference.

7. Measure, label, and organize the coordinates of M' and N'.

8. Adjust the value of k to equal 1, 2, and 3, by changing the size of the segment in the upper left hand portion of the screen. Record the coordinates of the image in the table below.

Magnitude (k) M M' N N'1 (1, 0.25) _________ (-0.50,0.50) _________2 (1, 0.25) _________ (-0.50,0.50) _________3 (1, 0.25) _________ (-0.50,0.50) _________

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9. Use the table to explain below how the coordinates of a point change under a transformation Sk.

Use k in your answer. Be precise.

10. Measure the length of segment MN and M'N'. Adjust the value of k by changing the size of the segment in the upper left hand portion of the screen, and compare the length of MN to its image M'N'. Explain below the relationship between the length of a segment and its image under a transformation Sk.

11. Measure the slope of segment MN and M'N'. Adjust the value of k and compare the slope of MN to its image M'N'. Explain below the relationship between the slope of a segment and its image under a transformation Sk.

10. Use your conclusions in this exploration to complete the missing information in the table below. Use the calculator to check your answers. Magnit.

kCoord.

MCoord.

M'Coord.

NCoord.

N'Length of MN

Length of M'N'

Slope of MN

Slope of M'N'

1/2 (1.00,.25) (-.5,.5) 1.52 -0.17(1.00,.25) (2.00,.50) (0,2) 1.26 -0.75

4 (4,0) (-.5,.5) 6.32 -0.33(2.25,3.0) (-.5,-.5) (-1.5,-1.5) 5.86 1.19

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EXPLORATION 12-2. SIZE CHANGES

Objective: Explore the transformation of a figure under a size change with varying magnitude.

1. CREATE A NEW DOCUMENT with the variable size. 2. Create a small horizontal segment in the upper left hand corner of the screen to represent the adjustable magnitude k. Measure the length of this segment and label the measurement with k=.

Create point B in the lower left corner of the screen to represent the center of dilation (or size change). Create a small quadrilateral QUAD near the left-middle of the screen with the Polygon tool, as shown to the right.

3. Use the Dilation tool to perform a size change of magnitude k on the quadrilateral about point B. You do not need to label the image Q'U'A'D' as shown to the right, but you must recognize which is the image of which.

4. Modify the value of k until the image Q'U'A'D' becomes smaller, the same size, and larger, than the preimage QUAD. Complete the blanks below.

The image contracts when ____ < k < ____The image remains the same size when k = ____The image expands when k > ____

5. Measure and label the distance from B to D, and B to D'. Modify the value of k and observe the changes in the measurements.

6. Predict and write an equation below using multiplication relating BD, B'D', and k, without using any numbers.

7. Use the Calculate tool and input the side of the formula with the k and BD. Modify the value of k and compare the formula's output with the measurement of B'D'. If the numbers are the same, then your formula is correct! If the numbers are not the same, check your formula with a classmate.

8. Measure and label the length of segments AD and A'D'. Modify the value of k and observe the changes in the measurements.

p. 93


9. Predict and write an equation below using multiplication relating AD, A'D', and k, without using any numbers.

10. Use the Calculate tool and input the side of the formula with the k and AD. Modify the value of k and compare the formula's output with the measurement of A'D'. If the numbers are the same, then your formula is correct! If the numbers are not the same, check your formula with a classmate.EXPLORATION 12-3. PROPERTIES OF SIZE CHANGES

Objective: Examine properties related to collinearity and angle measurement under a size change with varying magnitude.

1. OPEN the document with the variable size contained in your folder.

2. Delete all of the measurements and calculations, except for the value of k, as shown to the right.

3. Check the collinearity of the dilation point, a vertex, and its image. Press and choose 8:Check Property » 1:Collinear, move the pencil on top of point B until the screen reads THIS POINT, press , move the pencil on top of point D until the screen reads THIS POINT, press , move the pencil on top of point D' until the screen reads THIS POINT, and press .

A dotted box will appear on the screen, as shown to the right. Drag the box to the right side of the screen and press . The screen will tell you whether the points are collinear or not collinear.

4. Check the collinearity of the other three sets of points (BQQ', BAA', BUU'). Drag around the vertices of QUAD to see if the collinearity changes. Use your observations to describe a property of a size change below without using any form of the word collinear (or explain what collinearity means).

5. Delete the collinearity outputs on the screen. Measure and label all of the angles in QUAD and Q'U'A'D'. Drag around the vertices of QUAD and compare the corresponding angle measurements. Use your observations to describe another property of a size change below.

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EXPLORATION 12-4. PROPORTIONS

Objective: Use size changes to learn relationships about proportions.



3. Measure and label the lengths of the sides of both polygons.

4. Use the Calculate tool to compute the ratios , , , and

5. Modify the value of k and explain below how the ratios compare with each other and the magnitude value, k.

6. Proportions represent two equal ratios. For example, you can show that = . Use the Calculate tool to find at least two more proportions. Record those proportions below.

7. Use the Calculate tool to compute DQ*D'A' and DA*D'Q'. Modify the value of k and observe the changes in the calculations. Use the proportion in step 6 and explain below how these computations verify the Means-Extremes Property.

EXPLORATION 12-5. SIMILARITY

Objective: Verify properties of similarity through transformations and measurement.

p. 95



2. Delete all of the measurements and calculations, except for the lengths of the sides in QUAD, and the value of k, as shown to the right.

3. Adjust the value of k so that the image Q'U'A'D' is to the left of the center of the screen.

4. Create a vertical line to the right of Q'U'A'D' and Reflect Q'U'A'D' over the line.

Label the reflected polygon Q"U"A"D", and Hide Q'U'A'D', as shown the right.

5. Measure the corresponding angles, and corresponding proportions, , , and . Modify the value of k and observe the changes in measurement and calculation.

6. Explain below why QUAD and Q"U"A"D" are similar figures under a composite of a size change and reflection.

7. CREATE A NEW DOCUMENT with the variable similar.

8. Use perpendicular lines to construct right triangle ABC, with right angle B. Construct an altitude from B to hypotenuse AC, as shown to the right. Let the altitude intersect the hypotenuse at point D. Drag around the vertices of the triangle to ensure the right angle and altitude keep their form.9. Use the Measure tool to find similar triangles in the diagram (some are overlapping). State and draw below at least two pairs of similar triangles from the diagram. Justify why the triangles are similar.

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EXPLORATION 12-6. THE FUNDAMENTAL THEOREM OF SIMILARITY

Objective: Show the relationship between the magnitude of a size change (k) compared to the perimeter and area of two similar figures.



3. Measure the perimeters of the two quadrilaterals.

4. Predict and write an equation below using multiplication relating the perimeter of QUAD, the perimeter of Q"U"A"D", and k, without using any numbers.

5. Use the Calculate tool and input the side of the formula with the k and the perimeter of QUAD from your answer in #4. Modify the value of k and compare the formula's output with the measurement of the perimeter of Q"U"A"D". If the numbers are the same, then your formula is correct! If the numbers are not the same, check your formula with a classmate.

6. Measure the areas of the two quadrilaterals.

7. Predict and write an equation below using multiplication relating the area of QUAD, the area of Q"U"A"D", and k, without using any numbers. Hint: Area uses square units.

8. Use the Calculate tool and input the side of the formula with the k and the area of QUAD. Modify the value of k and compare the formula's output with the measurement of the area of Q"U"A"D". If the numbers are the same, then your formula is correct! If the numbers are not the same, check your formula with a classmate.

EXPLORATION 13-1. THE SSS SIMILARITY THEOREM

Objective: Show that three sets of equal ratios will produce similar triangles

1. CREATE A NEW DOCUMENT with the variable simsss.

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2. Create overlapping triangles CAT and FAR with common vertex A and segment FR parallel to segment CT, as shown to the right. You will have to construct a parallel line FR, then make an overlapping segment FR. Hide the parallel line (but not the segment) that is necessary in the construction.

Drag around the vertices of ∆CAT to ensure that FR remains parallel to CT.

3. Measure and label the lengths of all of the sides of both triangles.4. Use the Calculate tool to compute the following ratios: , , and . Drag around point F or point R and observe the changes in your calculations. What can you conclude about these ratios?

5. Without measuring, explain below why the corresponding angles in ∆CAT are equal to those in ∆FAR.

6. Given that three corresponding ratios are equal, is this enough to verify that ∆FAR~∆CAT? Explain why or why not below.

EXPLORATION 13-2. THE AA AND SAS SIMILARITY THEOREMS

Objective: Show that the AA and SAS conditions produce similar triangles

1. CREATE A NEW DOCUMENT with the variable simsas.

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2. Construct triangles ABC and EBD by creating two parallel lines and two intersecting segments between them, as shown to the right.

3. Create overlapping segments AC and DE. Hide the two lines.

4. Without measuring, state below three pairs of congruent corresponding angles. Explain why the pairs of angles are congruent.

5. Measure and label the lengths of the sides in both triangles. Use the Calculation tool to compute , , and .

6. Drag the vertices of the triangles around and observe the changes in ratios. Given that three corresponding angles are congruent (AAA), is this enough to verify that ∆ABC~∆EBD? Explain why or why not below.

7. Is it possible to show ∆ABC~∆EBD with only two pairs of congruent corresponding angles (AA)? Explain why or why not below.

8. Clear the screen. Construct two dotted concentric circles (same center) with center A. Create two non-overlapping isosceles triangles ABC and ADE, whose legs are equal to the radii of their respective circles, as shown to the right.

9. Measure and label < DAE and < CAB. Drag point E around the circle until both angles have the same measurement.

10. Without measuring, explain below why = .11. Measure and label DE, CB, EDA, DEA, ACE, and ABC. Use the Calculate tool to compute .

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12. Modify the larger circle's size and observe the changes in corresponding ratios and angle measurements. Given that two corresponding ratios are equal and an included pair of angle is congruent (SAS), is this enough to verify that ∆ABC~∆ADE? Explain why or why not below.

EXPLORATION 13-2P. TRIANGLES IN A TRAPEZOID

Objective: Find the relationship between the area of triangles in a trapezoid.

1. CREATE A NEW DOCUMENT with the variable tritrap.

2. Construct a trapezoid, diagonals, andintersection of diagonals as shown to the right.

3. Which triangles in the diagram are similar?Explain how you know.

4. Find an equation relating the ratio of the bases of the trapezoid, the area of triangle CED and the area of triangle AEB. Explain how you determined this result.

5. Which triangles in the diagram have the same area? Explain why this is true.

EXPLORATION 13-3. THE SIDE SPLITTING THEOREM

Objective: Verify the side-splitting theorem and its converse

1. OPEN the file simsss in your folder.

2. Delete all of the measurements and calculations, except for the lengths of FA and RA, as shown to the right.

3. Measure the lengths of segments FC and RT. Use the Calculate tool to compute the ratios and .

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4. Drag around point F or point R and observe changes in the calculated ratios.

5. If a parallel segment (AG) splits two sides of a triangle, what can you conclude about the ratios of the split sides? Explain below.

6. CREATE A NEW DOCUMENT with the variable sidespl.

7. Create overlapping triangles LUH and AUG, as shown to the right. Segment AG should not be drawn parallel to segment LH. Create an overlapping segment LH.

8. Measure and label the slopes of segments AG and LH, and the lengths of segments UA, AL, UG, and GH.

9. Use the Calculate tool and compute the ratios and .10. Drag point A or point G until = . Observe the changes in the slope of AG and LH. If a segment (AG) splits two sides of a triangle into equal proportions, what can you conclude about that segment and the base of the triangle (LH)? Explain why below.

INDEX

Operation

AA similarityAAS postulateacute trianglealternate interior anglesaltitude

trapezoidtriangle

angle(s)alternate interiorbisectorcentralcorrespondingcreateexteriorinteriornegativesize change

prop

First seen in Exploration

13-27-28-45-4

8-58-4

5-43-18-73-60-97-97-94-512-37-9

Operation

base of trianglebisector

angleperpendicular

calculatecentral anglechange anglecheck propertycircle

area (formula)area (measure)circumferenceconcentricdiameterequationradius

circumferenceformula


8-4

3-13-80-108-70-92-8

8-80-80-813-28-711-70-8

8-7p. 101


straightsum in polygonunequalvertical

arcarc lengtharea

circle (formula)circle (measure)irregular polylateralparallelogramrectanglesurfacetrapezoidtriangle (form)triangle (meas)

ASA postulateaxes

coordinateunit

5-77-93-36-18-7

8-80-8

8-3, 8-510-68-58-210-68-58-40-107-2

1-31-3

measureclear screencollinearitycommentcondition, SSAcongruent trianglescontrastcollecting datacommentconcentric circlescongruent trianglescoordinate

measureplane

corresponding anglescrossdarken the screendata collectiondeletediagonal

kiteparallelogram

diameterdilationdistance

formulameasureminimum

0-80-612-30-87-57-20-18-80-813-27-2

11-64-23-60-40-18-8

0-6, 0-8

6-47-78-712-1

11-60-84-3

INDEX (page 2)

Operation

document, newdotteddrawing, perspectiveedit

labelnumerical

encasementequation

circlelineregression

equilateral triangle


0-44-51-5

0-57-68-5

11-71-38-86-2

Operation

glide reflectiongridheight

trapezoidtriangle

hideHL postulateinterior angleintersecting linesintersection pointirregular polygon, areaisosceles trapezoid


4-71-3

8-58-41-57-57-90-70-7

8-3, 8-56-5

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errordeletewrong key

explicit formulaexterior anglesfolder, newformatformula

arc lengtharea, similarityarea of circlearea of rectanglearea of trianglecircumferencedistanceequat of circleequation of lineexplicitlateral arealine, equationmidpointperim,

similarityPythag Thmrecursiveslopesum of poly <sreg poly

rotationreg poly perimsurface areavolume

fund theorem similarity

0-60-25-77-90-31-3

8-712-68-88-28-48-711-611-71-35-710-61-311-812-68-65-73-65-76-68-110-610-612-6

isosceles trianglek, magnitudekitelabel

linemeasurementpoint

lateral area formulaslegs, isos. trapezoidlighten the screenline(s)

createequationhorizontallabelobliqueparallelperpendicularvertical

macromagnitudemeans-extremesmeasure

angleareacircumferencecoordinatesdistance lengthperimeterradiussegment lengthslope

median

6-212-16-4

3-60-80-510-66-50-1

0-71-31-33-60-7

1-5, 3-63-71-34-712-112-4

0-90-80-811-60-80-80-100-80-83-66-2

INDEX (page 3)

Operation

midlinemidpoint

connectioncreate


11-8

11-82-4

Operation

pointcreatingintersectionlabel


0-50-70-5

p. 103


formulaminimum distancemove

anglelabel measurementpoint

mystery quadrilateralsnegative anglenew

document/sketchfolderprogram

numerical editoblique linesobtuse triangleorthocenterparallel linesparallelogram

areaconstructdiagonalsproperties

perimeterregular polygontriangle

perpendicular bisectorbisector

theoremlines

perspective drawingplane, coordinate

11-84-3

0-90-50-90-56-M4-5

0-40-310-6

3-2, 6-38-48-4

1-5, 3-6

8-56-37-77-7

8-10-10

3-85-53-71-51-3

midpointmoveon objecton perp.

bisectorvanishing

pointerpolygon

createspecial quad

postulateAASASAHLSASSsASSS

press the wrong keyprogram, newproperties, size changeproportionsPythagorean Theoremquadrilateral

createmystery

radiusratiorayrectangle

areaconstructpolygon tool

recursive formula

2-40-50-95-51-50-4

4-78-2

7-27-27-57-27-57-20-210-612-312-48-6

6-36-M0-812-40-9

8-26-38-25-7

p. 104


INDEX (page 4)

Operation

reflectapplicationfigureglidepointsymmetric

regression equationregular polygon

createperimeter

remove object(s)revise a labelrhombusright trianglerotations of a figure

symmetrySAS postulateSAS similarityscalene trianglesegmentselect objectshowside splitting theoremsides, unequalsimilarity

AAfund theoremintroductionSASSSS

size changecreateproperties

Sk, transformationslidesslope

formulaidentifymeasure

split screensquare


4-34-44-74-16-18-8

1-5, 6-78-10-60-56-38-4

3-2, 4-56-67-213-26-20-8

0-6, 4-78-413-37-9

13-212-612-513-213-1

12-212-312-14-6

3-61-33-68-8

1-5, 6-3

Operation

sum of anglesformulatrianglepolygon

surface area formulassymmetry

reflectionrotation

straight angleswimmersysdatatessellationstransformation Sktranslationstransversaltrapezoid

areaconstruct

triangle(s)acutealtitudearea (formula)area (measure)basecongruentcreatingequilateralheightinequalityisoscelesobtuseorthocenterrightscalenetessellations

translationtransversalunequal angles/sidesunit, axesvanishing pointvectorvertical angles


5-76-15-710-6

6-16-67-94-38-87-612-14-63-6

8-56-5

8-48-48-40-108-47-20-106-28-42-76-28-48-48-46-27-64-63-67-91-31-54-63-3

p. 105


SSA conditionSsA postulateSSS postulateSSS similarity

7-57-57-213-1

volume formulaswrong key errory-intercept

10-60-21-3

VARIABLE INDEXVariable Name

altintangbisanglesareacircbouncecapturecirccircumfcomprefcompref2congrucongtriconjectcoordindistformeqcirceqlineextanggliderefinequalirrareaisostrikitelinesmidareamidarea2midpointmnmidmystery (folder)optimaloptimal2

Exploration

5-43-10-98-87-7P5-5P0-88-74-44-55-27-22-84-211-611-71-37-94-72-78-36-26-40-711-8P11-8P2-411-86-M8-2P8-2P

Topic

Alternate interior anglesAngle bisectorCreate, measure, and change an angleArea of a circle through data analysisBouncing off the wallsCapture the flagCreate, measure, and change a circleCircumference and arc lengthTranslations: Composing reflections over parallel linesRotations: Composing reflections over intersecting linesCongruence and equalityCongruent trianglesTesting conjecturesReflecting figures in a coordinate planeDistance formulaEquations of circlesEquations of linesExterior anglesGlide reflectionsTriangle inequalityAreas of irregular polygonsIsosceles and equilateral trianglesConstructing a kiteCreate two intersecting linesMidpoints and areasMidpoints and areasMidpointsMeans and midpointsMystery quadrilateralsOptimal quadrilateralsOptimal quadrilaterals

Continued on next page

p. 106


VARIABLE INDEX (page 2)Variable Name

parallelpath1path2pararectperpbisperplineperspectpgrampointpolysumptperppythagrectareareflectrefsymregperimregpolyrotaterotsymsharksidesplsimilarsimsassimsss

size

Exploration

3-69-59-56-33-83-71-57-70-4, 0-55-75-58-68-24-16-18-16-73-26-66-2P13-312-513-213-113-312-212-312-412-512-6

Topic

Slope and parallel linesFamous curvesFamous curvesConstructing parallelogramsPerpendicular bisectorPerpendicular linesDrawing in perspectiveProperties of parallelogramsCreate, label, and move pointsSum of angles in a polygonPerpendicular bisector theoremPythagorean TheoremArea of a rectangleReflecting pointsReflection-symmetric figuresPerimeter of a regular polygonRegular polygonsRotations of varying magnitudes and signsRotation symmetryShark attackThe side splitting theoremSimilarityThe AA and SAS similarity theoremsThe SSS similarity theoremThe side splitting theoremSize changesProperties of size changesProportionsSimilarityThe fundamental theorem of similarity

p. 107


skthmssahlswimmertancirctancirc2tancirc3tesselltrapareatrapeztreasuretriangletriareatrirectritrapvectorvertangyour name (pgm)

12-17-54-38-8P8-8P8-8P7-68-56-53-8P0-108-48-4P13-2P4-63-310-6

The transformation SkSsA condition and HL congruenceReflections and minimum distanceArea of tangent circlesArea of tangent circlesArea of tangent circlesTessellationsArea of trapezoidsConstructing a trapezoidFind the hidden treasureCreate, measure, and change a triangleArea of a triangleTriangle in a rectangleTriangles in a trapezoidTranslations and vectorsVertical anglesCreating a toolbar to reference formulas

p. 108


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