Mathematics Florida Standards Similarity in Right 7-4

7-4 Similarity in RightTriangles

Mathematics Florida Standards

MAFS.912.G-SRT.2.5 Use... similarity criteria fortriangles to solve problems and to prove relationships ingeometric Hgures. Also MAFS.912.G-GPE.2.5

MP1,MP3,MP4

Objective To find and use relationsliips in similar right triangles

Getting Ready! ► X ' C 4

Analyze thesituation first.Think about howyou will match

.angles.

MATHEMATICALPRACTICES

Draw a diagonal of a rectangular piece of paperto form two right triangles. In one triangle,draw the altitude from the right angle to thehypotenuse. Number the angles as shown. Cutout the three triangles. How can you match theangles of the triangles to show that all threetriangles ore similar? Explain how you know thematching angles are congruent.

6//5

2 3

T!

In the Solve It, you looked at three similar right triangles. In this lesson, you will learnnew ways to think about the proportions tliat come from these similar triangles. Youbegan with three separate, nonoverlapping triangles in the Solve It. Now you \vill seethe two smaller right triangles fitting side-by-side to form the largest right triangle.

Essential Understanding Whenyou draw the altitude to the hypotenuse of aright triangle, you form three pairs of similar right triangles.

LessonVocabularygeometric mean

Theorem 7-3

TheoremThe altitude to the hypotenuse of a right triangle divides the triangle intotwo triangles that are similar to the original triangle and to each other.

If . . . CAABC is a right trianglewith right AACB, andCD is the altitude to the

hypotenuse

D

Then. . .AABC ~ AACD

AABC ~ ACBD

A A CD ~ ACBD

460 Chapter 7 Similarity

\

Proof Proof of Theorem 7-3

Given: Right Ay4BCwith right AACB

and altitude CD

Prove: AACD ~ AABC, ACBD ~ AABC, AACD ~ ACBD

Statements Reasons

1) /-ACB is a right angle. 1) Given

2) CD is an altitude. 2) Given

3) W LAB 3) Definition of altitude

4) AADC and ACDB are right angles. 4) Definition of L

5) AADC = AACB, 5) All right A are =.

ACDB = AACB

6) AA = LA, LB = LB 6) Reflexive Property of =

7) A^CD - AABC, 7) AA ~ Postulate

ACBD ~ AABC

8) LACD = LB 8) Corresponding A of ~ As are =.

9) LADC = LCDB 9) All right A are =.

10) AACD - ACBD 10) AA ~ Postulate

What will helpyou see thecorrespondingvertices?

Sketch the trianglesseparately in the sameorientation.

Problem 1 Identifying Similar Triangles

What similarity statement can you write relating the three

triangles in the diagram?

YW is the altitude to the hypotenuse of right AXYZ, so you can use

Theorem 7-3. There are three similar triangles.

AXYZ ~ AFWZ ~ AXWY

Got It? 1. a. What similarity statement can you write relatingthe three triangles in the diagram?

b. Reasoning From the similarity statement inpart (a), write two different proportions using

the ratiosp-

X

w

c PowerGeometry.com Lesson 7-4 Similarity in Right Triangles 461

How do you usethe definition of

geometric mean?Set up a proportionwith* in both means

positions. The numbers6 and 15 go into theextremes positions.

Proportions in which the means are equal occur frequently in geometry. For any twopositive numbers a and b, the geometric mean of a and b is the positive number x such

CS) 30

Finding the Geometric Mean

Multiple Choice What is the geometric mean of 6 and 15?

CE:= sVIo c^eVIo

Definition of geometric mean

Cross Products Property

Take the positive square root of each side.

Write in simplest radical form.

The geometric mean of 6 and 15 is sVTo. The correct answer is B.

Got It? 2. What is the geometric mean of 4 and 18?

X 15

:c2 = 90

*:= V90

x = 3VlO

In Got It 1 part (b), you used a pair of similar triangles to write a proportion with a

geometric mean.

A5QP ~ ASPP

short leg _ long legshort leg long leg

SP is the geometric meanof SO and SR.

This illustrates the first of two important corollaries of Theorem 7-3.

Corollary 1 to Theorem 7*3

CorollaryThe length of the altitude to the

hypotenuse of a right triangle is the

geometric mean of the lengths of the

segments of the hypotenuse.

If... Then...

6D-Q9.CD DB

Example

Segments ofhypotenuse

Altitude to

hypotenuse

You will prove Corollary 1 In Exercise 42.

462 Chapter? Similarity

Corollary 2 to Theorem 7-3

Corollary If..The altitude to the hypotenuse of a right

triangle separates the hypotenuse so that

the length of each leg of the triangle is

the geometric mean of the length of the

hypotenuse and the length of the segment

of the hypotenuse adjacent to the leg.

Example

V

r

KM

Hypotenuse.

Then ...

AC AD

AB ̂ CBCB DB

'Leg

Segment of hypotenuseadjacent to leg

You will prove Corollary 2 in Exercise 43.

How do you decidewhich corollaryto use?

If you are using orfinding an altitude,use Corollary t, If youare using or finding aleg or hypotenuse, useCorollary 2.

The corollaries to Theorem 7-3 give you ways to write proportions

using lengths in right triangles without thinking through

the similar triangles. To help remember these corollaries,

consider the diagram and these properties.

Corollary 1

u a

Corollary 2

h__^ h__Si ' £n

Problem 3 Using the Corollaries

Algebra What are the values ofa: and y?

IJsp Cnrnllary 2. \ 4 + 12 _ xX 4

= 64

Write a proportion.


\

\

n

1 _ X ̂ Use Corollary 1. ]y 12 ^y2 = 48

X = -s/M Take the positive square root. y =

x = S Simplify.

Got It? 3. what are the values of x and y?4

y

5

\ j y/

C PowerGeometry.com Lesson 7-4 Similarity in Right Triangles 463

Problem 4 Finding a Distance Q2)

Robotics You are preparing for a robotics

competition using the setup shown here.

Points >1, B, and C are located so that

AB = 20 in., and AB 1 BC. Point D

is located on AC so that BD ± AC

and DC = 9 in. You program the

robot to move from AtoD and

to pick up the plastic bottle at

D. How far does the robot

travel from A to Dl

You can't solve this

equation by takingthe square root. Whatdo you do?Write the quadraticequation in thestandard form

ax^ + bx + c = 0.Then solve by factoringor use the quadraticformula.

x + 9 20

X20

x^ + 9x = 400

+ 9x - 400 = 0

(x - 16)(x + 25) = 0

X — 16 = 0 or

X = 16 or

(x + 25) = 0

x= -25

Corollary 2


Subtract 400 from each side.

Factor.

Zero-Product Property

Solve for x.

Only the positive solution makes sense in this situation. The robot travels 16 in.

Got It? 4. From point D, the robot must turn right and move to point B to put thebottle in the recycling bin. How far does the robot travel from D to S?

Lesson Check

Do you know HOW?Find the geometric mean of each pair of numbers.

1. 4 and 9 2. 4 and 12

Use the figure to complete each proportion.

a f ^e~ m

5 ̂ = ̂

4d ■

MATHEMATICAL

Do you UNDERSTAND? PRACTICES

7. Vocabulary Identify the following in AftST.a. the hypotenuse ^b. the segments of the hypotenuse

c. the segment of the hypotenuse

adjacent to leg ST

8. Error Analysis A classmate

wrote an incorrect

proportion to find x. Explain

and correct the error.


Practice and Problem-Solving Exercises

,1^ Practice Write a similarity statement relating the three trian

MATHEMATICAL

PRACTICES

gles in each diagram.

9.

N.

K

11.M

y

^ See Problem 1.

N

0

Algebra Find the geometric mean of each pair of numbers.

12. 4 and 10 13. 3 and 48

15. 7 and 9 16. 3 and 16

^ See Problem 2.

14. Sand 125

17. 4 and 49

Algebra Solve for A:andy.

20.

22. Architecture Hie architect's side-view drawing of

a saltbox-style house shows a post that supports the roof ridge.The support post is 10 ft tall. How far from the front of the

house is the support post positioned?

13 Apply @23.8. The altitude to the hypotenuse of a right triangle dividesthe hypotenuse into segments 2 cm and 8 cm long. Find

the length of the altitude to the hypotenuse.

b. Use a ruler to make an accurate drawing of the right

triangle in part (a).c. Writing Describe how you drew the triangle in part (b).

Algebra Find the geometric mean of each pair of numbers.

24. 1 and 1000 25. 5 and 1.25 26. and V2

^ See Problems 3 and 4.

21. y

Supportpost

Bedroom

Living room Kitchen

27. 2 and 2 28. and V?

29. Reasoning A classmate says the following statement is true: The geometric mean

of positive numbers a and b is V^. Do you agree? Explain.

30. Think About a Plan The altitude to the hypotenuse of a right triangle divides thehypotenuse into segments with lengths in the ratio 1: 2. The length of the altitude

is 8. How long is the hypotenuse?

• How can you use the given ratio to help you draw a sketch of the triangle?

• How can you use the given ratio to write expressions for the lengths of the

segments of the hypotenuse?

• Which corollary to Theorem 7-3 applies to this situation?

C PowerGeometry.com Lesson 7-4 Similarity in Right Triangles 465

31. Archaeology To estimate the height of a stone figure, Anyaholds a small square up to her eyes and walks backward from

the figure. She stops when the bottom of the figure aligns with

the bottom edge of the square and the top of the figure aligns

with the top edge of the square. Her eye level is 1.84 m from the

ground. She is 3.50 m from the figure. What is the height of the

figure to the nearest hundredth of a meter?

32. Reasoning Suppose the altitude to the hypotenuse of a righttriangle bisects the hypotenuse. How does the length of the

altitude compare with the lengths of the segments of the

hypotenuse? Explain.

The diagram shows the parts of a right triangle with

an altitude to the hypotenuse. For the two given

measures, find the other four. Use simplest

radical form.

33. = 2, S] = 1 34. = 6, Sj = 6 35. = 2,52 = 3 36. Sj = 3, ̂2 = 6V3

37. Coordinate Geometry CD is the altitude to the hypotenuse of right AABC.The coordinates of A, D, and B are (4,2), (4,6), and (4,15), respectively. Find all

possible coordinates of point C.

Algebra Find the value of a:.

X + 3

12

x + 2

Use the Rgure at the right for Exercises 42-43.

42. Prove Corollary 1 to Theorem 7-3.

Given; Right AABC with altitude

to the hypotenuse CD

Prove- ̂ = ̂rrove.

Prove Corollary 2 to Theorem 7-3.

Given: Right AABC with altitude

to the hypotenuse CD

Prove- =AC AD'EC DB

44. Given: Right AABC with altitude CD to the hypotenuse ABProof

Prove: The product of the slopes of perpendicular lines is — 1.

Challenge 45. a. Consider the following conjecture: The product of thelengths of the two legs of a right triangle is equal to the

product of the lengths of the hypotenuse and the altitude

to the hypotenuse. Draw a figure for the conjecture. Write

the Given information and what you are to Prove.

b. Reasoning Is the conjecture true? Explain.

s. y ysc

/6

A'-a- i X

0 1 nK


46. a. In the diagram, c = x + y. Use Corollary 2 to Theorem 7-3 to write two more

equations involving a, b, c, x, andy.

b. The equations in part (a) form a system of three equations in five variables.Reduce the system to one equation in three variables by eliminating x and y.

c. State in words what the one resulting equation tells you.

47. Given: In right AABC, BD 1 AC, and DE J_ BC.Proof Prove- ̂ = ̂rrove.

BE

EC

B C

^^/Aa

Short

Response

Standardized Test Prep

48. The altitude to the hypotenuse of a right triangle divides the hypotenuse into

segments of lengths 5 and 15. What is the length of the altitude?

Ca:)3 C1>5V^ CO 10 C0 5V^

49. A triangle has side lengths 3 in., 4 in., and 6 in. The longest side of a similar triangleis 15 in. What is the length of the shortest side of the similar triangle?

CDlin. CO 1-2 in. CO 7.5 in. CD 10 in.

50. Two students disagree about the measures of angles in a kite. They know that twoangles measure 124 and 38. But they get different answers for the other two angles.Can they both be correct? Explain.

rMixed Review

51. Write a similarity statement for the two triangles.

How do you know they are similar?R

M'

R

N

V,Algebra Find the values of a: and y in oRSTY.

52. RP = 2x, PT = y 2, ICP = y, PS = x -I- 3

53. R17= 2x -H 3, VT= 5:c, rs =y -I- 5, Sfl = 4y - 1

Get Ready! To prepare for Lesson 7-5/ do Exercises 54-56.

The two triangles in each diagram are similar. Find the value of x in each.

^ See Lesson 7-3.

^ See Lesson 6-2.

30 cm

55.

12 in.

56.

^ See Lesson 7-2.

11 mm

6 mm\w 4 mm

C PowferGeometry.com Lesson 7-4 Similarity in Right Triangles 467

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Mathematics Florida Standards Similarity in Right 7-4