Rigid Motions Proof – Cheat Sheet Key Concepts Additional Notes

Refresher- List all the Rigid Motions we’ve learned:

Mapping FiguresThe idea of pairing each vertex of the preimage to its corresponding point on the image.

How we use mapping to prove congruence: If we can map every single point of the pre-image onto its corresponding points on the image (w/ rigid motions), then the figures are congruent.Proof TYPE 1: Describing the rigid motions that will result in congruent figures:Example: Describe what type of rigid motion(s) you would use to map ΔABC onto ΔDEF.

First, translate Triangle ABC along vector AD, such that A maps to D. Then, Rotate Triangle A’B’C’ around point A’ such that AC maps onto DF. Lastly, Reflect Triangle A’’B’’C’’ over A’’C’’, such that B’’ maps onto E’’.

Are the two triangles congruent? Explain your reasoning.Yes, these triangles are congruence since there is a sequence of rigid motions that allow all corresponding sides to be mapped onto one another.Proof TYPE 2: Two Column Rigid Motion Proof

Fact Check: In an isosceles triangle, if a perpendicular bisector is drawn in, it is also the ______________________________ of the vertex angle.

So, If we are given : 1. BD is Angle bisector in an isosceles triangleThen we know: 2. _________________________________ Which means we also have a 3. _________________________________

Important! YOU MUST INCLUDE THIS WHEN DESCRBING A:

1. Reflection -> line/segment of reflection.

2. Rotation-> Around what point? Orientation? Where do you stop rotating? So that __ goes to __.

3. Translation -> along VECTOR!

How do we recognize them? Look for trigger words!

IMAGE LINE OF REFLECTION REFLECTED OVER TRANSFORMATIONS RIGID MOTIONS

Highlight important details as we go through this!!

Example: Seen below!Regents-Style Questions

1. Given: Isosceles Δ ABC ,BA≃BC and BD , the angle bisector of ∠B

Prove: Δ ABD≃ΔCBD

Statements Reasons

1. Isosceles Δ ABC ,BA≃BC and BD , the angle bisector of ∠B

1. Given

2. BD is the perpendicular bisector 2. In Isosceles Triangles, the perpendicular bisectors and angle bisectors coincide.

3. BD is the line of reflections 3. Perpendicular bisectors are lines of reflection

4. AD ≅ CD 4. Definition of perpendicular bisector

5. A maps onto C 5. AD ≅ CD and bd is the line of reflection

6. B maps onto B 6. B is on the lines of reflection

7. D maps onto D 7. D is on the lines of reflection

8. Δ ABD≃ΔCBD 8. All corresponding vertices of the preimage and image can me mapped onto each other through rigid motions.

Try Another! In the diagram of and below, , , and .

Highlight important details as we go through this!!

a) Prove that . Why is this a shortcut proof?

b) Describe a sequence of rigid motions that will map onto .

Right Triangle Trig – Cheat SheetKey Ideas Additional NotesRadicals: -Rationalize Denominators

Simplify 2√7

⋆ √7√7

=2√77

Trigonometric Ratios – SOHCAHTOA ****PUT CALCULATOR IN DEGREES*** Used to solve for sides

or to solve for angles

When solving for an angle

remember to use: sine,

cosine, or tangent

*Angle of Elevation – angle

formed from ground up

*Angle of Depression – angle formed from eye-level down

Auxiliary lines to solve in non-right trianglesTips:

Read carefully, if it is NOT a right triangle we can’t use SOHCAHTOA!

Auxiliary lines will always be an ALTITUDE to help us get right triangles so we CAN use SOHCAHTOA

*NOTE: An altitude is not always an angle bisector

Try it! See the sketched auxiliary line to help create right triangles in the diagram:

Highlight important details as we go through this!!

Cofunctions

sin θ=¿cos¿¿

cosθ=¿ sin ¿¿

Try it! Write a trig function equivalent to the following, but with an angle value less than 45°

sin 78 ¿cos¿ = cos (12)

Determine Exact values Trig Values in Special Right Triangles

PRACTICE !

45-45-90 30-60-90

Trigonometric Identities

Try it! Simplify the following trig expressions:

a)

tanθsin θ =

sin θcosθsin θ =

sin θcosθ

∙ 1sinθ=

1cosθ

Try it!

(cos 60°)(sin 45°)

tanθ= sin θcosθ

Highlight important details as we go through this!!

Regents-Style Questions

1. In , the complement of is . Which statement is always true?

1)

2)

3)

4)

2. Take a look at a Student Sample…

**It pays to show your work, even the set-ups!!

Let’s Try One!! A flagpole casts a shadow 16.60 meters long. Tim stands at a distance of 12.45 meters from the base of the flagpole, such that the end of Tim's shadow meets the end of the flagpole's shadow. If Tim is 1.65 meters tall, determine and state the height of the flagpole to the nearest tenth of a meter.

Area on the Plane - Cheat SheetKey Topics Additional NotesGraphing inequalities:

Describing regions using inequalities:

Be careful with vertical and horizontal lines!

Watch out with the direction of shading!

Using inequalities, describe the polygonal region shown below:

Area of Overlapping polygons

Highlight important details as we go through this!!

Area of Overlapping Regions

Remember! Union (∪¿- Add area of each shape minus overlap

Intersection¿)- Area of JUST the overlap

Area of Slanted Polygons Draw a “box” around slanted polygon. Label each region with Roman numeral (including the missing region). Calculate the area of each region and subtract form the area of the “box”

Approximating the area of curved Regions:Approximate area of curved Regions

Remember! Under Approx (stay UNDER the line or inside shape)

Over Approx ( stay over the line or outside of shape)

Remember: The SMALLER the boxes we use inside the curved region, the BETTER the approximation!

Take the

Average!

Example:

Finding the shortest distance between a point and a line.

Memorize!

Review Session 3 Practice Problems1. As graphed on the set of axes below, is the image of after a sequence of transformations.

Is congruent to ? Use the properties of rigid motion to explain your answer.

2. Describe a sequence of rigid motions that will map ∆ ABC ¿∆≝¿

3. Given: G is the image of E after a reflection over DF and ∆≝¿∆DGF are drawn

Prove ΔDEF≃ΔDGF using rigid motions

Statements Reasons

1. 1. Given

2. 2.

3. 3.

4. 4.

4. a) List a sequence of rigid motions that will map ∆ JKH ¿∆FGD . (You don’t have to describe here)

b) Use the properties of rigid motions to explain why ∆ JKH ≅ ∆ FGD

5. Use the diagram shown where ∆LAC≅ ∆DNC for parts A and B.

a)Fill in:

Mapping:

The rigid motion RC D maps onto __

The rigid motion RC N maps onto __

The rigid motion RC C maps onto __

6. In the diagram of ΔLAC and ΔDNC below, LA≃DN , CA≃CN , andDAC⊥ LCN .

. Describe a rigid motion(s) that will map ΔLAC onto ΔDNC .

7. Given: C is the image of A after a reflection over BD ∆ ABD∧∆CBDare drawn.

Prove using rigid motions: Δ ABD≃ΔCBD

Complete the proof: Δ ABD≃ΔCBD

Statements Reasons

1. : C is the image of A after a

reflection over BD and Δ ABDand ΔCBD are drawn.

1. Given

2. 2.

3. 3.

4. Δ ABD≃ΔCBD 4.

8. In the diagram below, and points A, C, D, and F are collinear on line .

Let (not drawn) be the image of after a translation along , such that point D is mapped onto point A.

What will the next rigid motion be in this sequence to map ∆D’E’F’ onto ∆ABC?

Are these triangles congruent? Why or why not?

9. In scalene triangle ABC shown in the diagram below, .

Which equation is always true?

1) Sin A = Sin B2) Cos A = Cos B3) Cos A = Sin C4) Sin A = Cos B

10. In a right triangle ABC, with right angle C, Sin A = Cos 9A. Using this, A is equal to…

11. If Sin(x+20) = Cos(x), find the value of x.

12. In the diagram of right triangle ABC shown below, and .

What is the measure of , to the nearest degree?

13. A carpenter leans an extension ladder against a house to reach the bottom of a window 30 feet above the ground. As shown in the diagram below, the ladder makes a 70° angle with the ground. To the nearest foot, determine and state the length of the ladder.

14. As shown in the diagram, the angle of elevation from a point on the ground to the top of the tree is 34°. If the point is 20 feet from the base of the tree, what is the height of the tree, to the

nearest tenth of a foot?

15. This is a toughy – but was a regents question!! Cathy wants to determine the height of the flagpole shown in the diagram below. She uses a survey instrument to measure the angle of elevation to the top of the flagpole, and determines it to be 34.9°. She walks 8 meters closer and determines the new measure of the angle of elevation to be 52.8°. At each measurement, the survey instrument is 1.7 meters above the ground.

Determine and state, to the nearest tenth of a meter, the height of the flagpole.

16. The diagram below shows a ramp connecting the ground to a loading platform 4.5 feet above the ground. The ramp measures 11.75 feet from the ground to the top of the loading platform.

Determine and state, to the nearest degree, the angle of elevation formed by the ramp and the ground.

17. As shown in the diagram below, a ship is heading directly toward a lighthouse whose beacon is 125 feet above sea level. At the first sighting, point A, the angle of elevation from the ship to the light was 7°. A short time later, at point D, the angle of elevation was 16°.

To the nearest foot, determine and state how far the ship traveled from point A to point D.

18. In the accompanying diagram of right triangles ABD and DBC, , , and . Find the

length of , leave answer in simplest radical form.

19. Rationalize the following:

A) 14√7

B)√8√2

20. Which ratio represents the cosine of angle A in the right triangle below?

a) 35 b)5

3 c)45 d)4

3

Know the difference! Solving for an angle and solving for a side!!

21.

a) A tree casts a 25-foot shadow on a sunny day, as shown in the diagram below

If the angle of elevation from the tip of the shadow to the top of the tree is 32°, what is the height of the tree to the nearest tenth of a foot?

21. Find the exact value of:

A) (sin45)(cos45)(tan60)

B) cos30sin 45

b) The diagram below shows the path a bird

flies from the top of a 9.5 foot tall sunflower to a

point on the ground 5 feet from the base of the

sunflower.

To the nearest tenth of a degree what is the

measure of angle x?

22. Simplify each expression:

a) 1−cos2θ

b ) tanθ cosθ

c )

tanθ cosθsin θ

23 .Given triangle ¿, ∠D=22 °, ∠F=91 °, DF=16.55, and EF=6.74,a) find DE to the nearest hundredth.

b) Solve for the area of triangle DEF. (Hint: You need the altitude).

24. Graph the solution set for the inequality on the set of axes below. Determine if the point is in the solution set. Justify your answer.

25. Using inequalities, describe the polygonal region shown below:

26. Using inequalities, describe the region with a system of inequalities. (Note: here it is not an enclosed figure)!

27. a) Determine the region that satisfies the following system of linear inequalities: y ≤ xy ≥ 2x < 5

b) What polygonal region is formed by the solution set?

c) What is the area of that polygon?

28. Determine the approximate area of the following curved region.

29. You are asked to approximate the area of the US. After copying a replica of the outline of the US using a scale, you realize in your desk you only have two types of graph paper to use! Which piece of graph paper would get a better estimated value for the area of the US? Justify your answer!

30. In the accompanying diagram, right triangle ABC is inscribed in a circle. BA is the diameter, BC =6 cm, AC = 8 cm. Find the exact area of the shaded region. Leave your answer in terms of π

31. Polygon A is a square and Figure B is a circle. Calculate the area of A U B. Round your answer to the nearest inch.

32. Determine the area of the following quadrilateral

33. Determine the area of a triangle ABC, whose vertices are A(-3,1) , B(1,3) and C(3,-2).

34.

Determine the shortest distance between the line and the given point, leave answer in radical form.

35. Determine the distance between a point and a line: y=−12x+2 and point (4,-2) round to the

nearest whole number.

Hi Geometry Friend! You worked hard! Check your work to see if you did it

right!