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460 Chapter 7 Similarity
Similarity in Right
Triangles
Objective To find and use relationships in similar right triangles
Draw a diagonal of a rectangular piece of paper to form two right triangles. In one triangle, draw the altitude from the right angle to the hypotenuse. Number the angles as shown. Cut out the three triangles. How can you match the angles of the triangles to show that all three triangles are similar? Explain how you know the matching angles are congruent.
In the Solve It, you looked at three similar right triangles. In this lesson, you will learn new ways to think about the proportions that come from these similar triangles. You began with three separate, nonoverlapping triangles in the Solve It. Now you will see the two smaller right triangles fitting side-by-side to form the largest right triangle.
Essential Understanding When you draw the altitude to the hypotenuse of a
right triangle, you form three pairs of similar right triangles.
Theorem 7-
Theorem The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other. If... N ABC is a right triangle with right ACB , and CD is the altitude to the hypotenuse
Then... N ABC N ACD N ABC N CBD N ACD N CBD A
A
D
C C B
D D
C
B
Analyze the situation first. Think about how you will match angles.
Dynamic Activity Similarity in Right TrianglesT
ACT IVITIES
AAAAAAAA AA CC CC D I E SSSSSSSS D S
YNAMIC
Lesson Vocabulary tgeometric mean
L V
L V tg
LL VVV t g
Content Standards
G.SRT.5 Use... similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.
MATHEMATICAL PRACTICES
Problem 1
Got It?
Lesson 7-4 Similarity in Right Triangles 461
Proof of Theorem 7- Given: Right N ABC with right ACB and altitude CD Prove: N ACD N ABC , N CBD N ABC , N ACD N CBD
Statements Reasons
ACB is a right angle. 1) Given
CD is an altitude. 2) Given
CD AB 3) Definition of altitude
ADC and CDB are right angles. 4) Definition of
ADC ACB , CDB ACB
All right are .
A A , B B 6) Reflexive Property of
N ACD N ABC , N CBD N ABC
AA Postulate
ACD B 8) Corresponding of are .
ADC CDB 9) All right are .
N ACD N CBD 10) AA Postulate
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 N XYZ , so you can use Theorem 7-3. There are three similar triangles.
N XYZ N YWZ N XWY
1. a. What similarity statement can you write relating the three triangles in the diagram? b. Reasoning From the similarity statement in part (a), write two different proportions using the ratio SRSP.
Proof
A D
C
B
W X
Z Y
W
X
Z Y
X
X
Z Y
Y
Z WY W
Q R
P
S
t
Th
What will help you see the corresponding vertices? Sketch the triangles separately in the same orientation.
Problem 3
Got It?
Lesson 7-4 Similarity in Right Triangles 463
Th e 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 Corollary 2 s 1 a^5
a s 2
h / 1 5
/ 1 s 1 ,^
h / 2 5
/ 2 s 2
Using the Corollaries Algebra What are the values of x and y?
3. What are the values of x and y?
1 2
s 1 s 2 a
h
x 2 64
x 8
x 64
4 y
y 12 y 2 48
y (^4) 3
y 48
Cross Products Property
4 x 12 x 4 Write a proportion.
Take the positive square root. Simplify.
Use Corollary 2. Use Corollary 1.
x
y
12
4
y x
4 5
Corollary 2 to Theorem 7-
Corollary Th e 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.
If... Then... AB AC^5
AC AD AB CB^5
CB DB
Example
You will prove Corollary 2 in Exercise 43.
A D
C
B
Hypotenuse
Segment of hypotenuse adjacent to leg
2
1 4
Leg
2
^2
How do you decide which corollary to use? If you are using or fi nding an altitude, use Corollary 1. If you are using or finding a leg or hypotenuse, use Corollary 2.
Problem 4
Got It?
464 Chapter 7 Similarity
Finding a Distance Robotics You are preparing for a robotics competition using the setup shown here. Points A, B, and C are located so that AB 20 in., and AB BC. Point D is located on AC so that BD AC and DC 9 in. You program the robot to move from A to D and to pick up the plastic bottle at D. How far does the robot travel from A to D?
x 9 20 ^
20 x Corollary 2 x^2 9 x 400 Cross Products Property x^2 9 x 400 0 Subtract 400 from each side. ( x 16)( x 25) 0 Factor. x 16 0 or ( x 25) 0 Zero-Product Property x 16 or x 25 Solve for x. Only the positive solution makes sense in this situation. The robot travels 16 in.
4. From point D , the robot must turn right and move to point B to put the bottle in the recycling bin. How far does the robot travel from D to B?
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.
3.
g e ^
e J 4.^
j d ^
d J
5. J f (^) J f 6.
j J ^
J g
Do you UNDERSTAND?
7. Vocabulary Identify the following in N RST. 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.
d f g h
e
j
S T
R P
Lesson Check
O
You can’t solve this equation by taking the square root. What do you do? Write the quadratic equation in the standard form ax^2 bx c 0. Then solve by factoring or use the quadratic formula.
e.
D
AC
m the D andd ottle aat robott D?
C D A
B
9 in.
20 in.
x
STEM
MATHEMATICAL PRACTICES
466 Chapter 7 Similarity
31. Archaeology To estimate the height of a stone figure, Anya holds 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 right triangle 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. h 2, s 1 1 34. a 6, s 1 6 35. 1 2, s 2 3 36. s 1 3, 2 6 3 37. Coordinate Geometry CD is the altitude to the hypotenuse of right N ABC. 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 x.
38. 39. 40. 41.
Use the figure at the right for Exercises 42–43.
42. Prove Corollary 1 to Theorem 7-3. Given: Right N ABC with altitude to the hypotenuse CD Prove: ADCD CDDB
FPO hsm11gmse_0704_a 12p0 w X 12p0 H
1 2
s 1 s 2 a
h
12 x
x 3
x
x 2
5
x (^18 )
x 20
x 5
A D
C
B
Proof
44. Given: Right N ABC with altitude CD to the hypotenuse AB Prove: The product of the slopes of perpendicular lines is 1. 45. a. Consider the following conjecture: The product of the lengths 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.
C
A B
a D c O
b
y
x
Proof
C^ Challenge
43. Prove Corollary 2 to Theorem 7-3. Given: Right N ABC with altitude to the hypotenuse CD Prove: ABAC ACAD , ABBC (^) DBBC
Proof
FPO hsm11gmse_0704_a 12p0 w X 12p0 H
3.50 m
1.84 m
Lesson 7-4 Similarity in Right Triangles 467
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 , and y. 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 N ABC, BD AC , and DE BC. Prove: ADDC (^) ECBE
a b x c
y
Proof B^
E
A D
C
Mixed Review
51. Write a similarity statement for the two triangles. How do you know they are similar?
Algebra Find the values of x and y in ^ RSTV.
52. RP 2 x , PT y 2, VP y , PS x 3 53. RV 2 x 3, VT 5 x , TS y 5, SR 4 y 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.
54. 55. 56.
R^ See Lesson 7-3. M (^) N
Q
P
V^ See Lesson 6-2.
R
P
T
S
See Lesson 7-2.
30 cm x (^) 15 cm
84 cm (^) 12 in. 5 in. 7 in. x
11 mm
6 mm 4 mm x
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? 3 5 3 10 5 5 49. A triangle has side lengths 3 in., 4 in., and 6 in. The longest side of a similar triangle is 15 in. What is the length of the shortest side of the similar triangle? 1 in. 1.2 in. 7.5 in. 10 in. 50. Two students disagree about the measures of angles in a kite. They know that two angles measure 124 and 38. But they get different answers for the other two angles. Can they both be correct? Explain.
SAT/ACT
Short Response
