HW problem 1: Use Geometer’s Sketchpad to construct a rectangle whose side lengths are in the ratio of 2:1 without using the perpendicular, parallel, or midpoint options in the construct menu, and without constructing any circles.

HW Problem 3: Construct a line segment AB using Geometer’s Sketchpad. Without changing line segment AB in any way, construct a rhombus (not a square) so that AB is a diagonal of the rhombus.

HW Problem 2:

Construct the letter A using Geometer’s Sketchpad.

Your A must be perfectly vertical and symmetric. In other words, it cannot look like this or this .

A

PD

A

C

B

4. Using Geometer’s Sketchpad, construct a regular pentagon with two diagonals, as shown below. 

a. Conjecture a relationship between the lengths of and . Prove your conjecture.   b. Compute and display the ratio of BP to AP. Compute and display the ratio of AB to AD. Compute and display the ratio of AD to AP.   c. Conjecture a relationship between the three ratios found in part (b).

d. Is there any special significance to the value of this ratio? Explain.

AC CP

Conjecture: All three ratios are equal.

C

B

A

5. A regular hexagon and a regular pentagon are given in the diagram. What is the measure of ABC? It is not necessary to use Geometer’s Sketchpad for this question.

132

6. Shown is a drawing of a kite. The drawing is rotated 180 around point P and the rotation image is then reflected over line m. Which of the following represents the image after the reflection?

A. B.

C. D.

P m

m P

m P

m P

m P

7. Recall problem 3 from Homework # 6:

“In the diagram, CDAB , mADC = 63 , mDCA = 41 , and mACB = 104 . Compute the measure of ABC.” Part of the solution was to “Flip triangle ABC so that points C and A are reversed.” Use Geometer’s Sketchpad to construct the diagram (with the correct angle measures) and then verify the solution by “flipping” triangle ABC so that points C and A are reversed and verifying that ABC DAC.

A

C

B

D

Do NOT email your construction to me. Be prepared to share your method of construction with the class next time.

8. A certain regular polygon has 90 diagonals. How many sides does the polygon have?  9. A certain regular polygon has angles that measure 165 each. How many sides does the polygon have?

10. In the polygon shown, sides AB, BC and CD are sides of a regular octagon,

and sides DE, EF, FG, and GA are sides of a regular decagon. Compute the measure of angle BAG.

A

B C

D

E

F

G

15 sides

24 sides

99

Definition:Two triangles are similar if three angles of one are congruent to three angles of the other, and their corresponding sides are in proportion.

D

E

F

C

B

A

ACDF

BCEF

ABED

DEF ABC

The ratio of any pair of corresponding sides is called the Scale Factor.

9

12

scale factor 3 : 4

Definition:Two triangles are similar if three angles of one are congruent to three angles of the other, and their corresponding sides are in proportion.

Similar triangle theorems:

Two triangles are similar if two angles of one triangle are congruent to two angles of the other.

Two triangles are similar if two sides of one triangle are proportional to two sides of the other, and the angles between these sides are congruent.

Two triangles are similar if all three sides of both triangles are proportional.

D

E

F

C

B

A

ACDF

BCEF

ABED

AA~

SAS~

SSS~

Can the triangles be proven similar? If so, state the similarity theorem that supports the conclusion.

1. 2.

9 15

1220

14x

Yes, AA~ Yes, SAS~

What is the scale factor?

x =3

123

35

Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, it divides the other two sides proportionally.

ECAE

BDAD

E

A

B C

D

6

10

8

x

20

y

3

1137.5

BCDE

ACAE

ABAD

Because ADE ABC,

P

N

M

L

K

J

Three or more parallel lines divideall transversals proportionally.

NP

MN

KL

JK

a

b

=a

b

DilationsUse Geometer’s Sketchpad to construct an acute triangle ABC near the lower left of the screen.

B

AC

P

Construct a triangle similar to, but not congruent to ABC.

DilationsUse Geometer’s Sketchpad to construct an acute triangle ABC near the lower left of the screen.

B

AC

P

Construct a triangle similar to, but not congruent to ABC.

A dilation is a transformation that 1.preserves angle measure and 2.changes lengths proportionally.

When a figure is dilated using scale factor k, the image is k times as far from the center of the dilation as the original figure.

Another way to construct the midpoint of a segment

Question # 5 on the Final Group Problem Solving Project

1. What will happen to the height of the point where the wires cross if the poles are moved further apart or closer together?

KSU

1. What will happen to the height of the point where the wires cross if the poles are moved further apart or closer together?

2. How does the height of the point relate to the heights of the flagpoles?

KSU

(This is what problem V is all about.)

D

C

B

A

Use Geometer’s Sketchpad to construct right triangle ABC in which mABC = 30 and mACB = 60. Then construct point D so that AD = (AB). Make a conjecture about how CD and DB are related. 1

3

PRACTICE WITH DILATION:

Are there any similar triangles in this diagram? If so, name them.