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Activity Assess 9-6 EXPLORE & REASON Right and Consider △​ ABC​ with CD‾​​ ​​ as shown. the Pythagorean B D

PearsonRealize.com A 45 C 5√2 I CAN… prove the using A. What is the of △​ ABC​ ? ​Of △ACD​ ? Explain your answers. and establish the relationships in special right B. Find the of ​​AD‾ ​​ and AB‾​​ ​​. triangles. C. Look for Relationships Divide the of the of △​ ABC​ VOCABULARY by the length of one of its sides. Divide the length of the hypotenuse of ​ △ACD​ by the length of one of its sides. Make a conjecture that explains • the results.

ESSENTIAL QUESTION How are similarity in right triangles and the Pythagorean Theorem related?

Remember that the Pythagorean Theorem and its converse describe how the side lengths of right triangles are related.

THEOREM 9-8 Pythagorean Theorem

If a is a , If... ​△ABC​ is a right triangle. then the sum of the of the B lengths of the legs is equal to the of the length of the hypotenuse. c a

A C b

2 2 2 PROOF: SEE EXAMPLE 1. Then... a ​​ ​ + ​b​​ ​ = ​c​​ ​

THEOREM 9-9 Converse of the Pythagorean Theorem

2 2 2 If the sum of the squares of the If... a ​​ ​ + ​b​​ ​ = ​c​​ ​ lengths of two sides of a triangle is B equal to the square of the length of the third side, then the triangle is a right triangle. c a

A C b

PROOF: SEE EXERCISE 17. Then... ​△ABC​ is a right triangle.

452 TOPIC 9 Similarity and Right Triangles Go Online | PearsonRealize.com Activity Assess

PROOF EXAMPLE 1 Use Similarity to Prove the Pythagorean Theorem

Use right triangle similarity to write a proof of the Pythagorean Theorem.

Given: ​△XYZ​ is a right triangle. Y Prove: a ​​ 2​ ​b​​ 2​ ​c​​ 2​ + = c a Plan: To prove the Pythagorean Theorem, draw the altitude to the hypotenuse. Then use the relationships in X Z b the resulting similar right triangles. Proof:

Step 1 Draw altitude XW‾​​ ​​. Y e c W Step 2 ​△XYZ ∼ △WXZ ∼ △WYX​ a f d

LOOK FOR RELATIONSHIPS X Z Think about how you can apply b properties of similar triangles. Step 3 Because What is the relationship between _c _a ​△XYZ ∼ △WYX​, ​​ a ​ = ​ e ​. 2 corresponding sides of similar So ​a​​ ​ = ce​. triangles? Step 4 Because △​ XYZ ∼ △WXZ​, c b ​​ _ ​ _​ ​. So ​b​​ 2​ cf​. b = f =

Step 5 Write an that relates ​​a​​ 2​ and ​​b​​ 2​ to ce and cf. 2 2 a ​​ ​ + ​​b​​ ​ = ce + cf 2 2 a ​​ ​ + ​​b​​ ​ = c(e + f) 2 2 a ​​ ​ + ​​b​​ ​ = c(c) 2 2 2 a ​​ ​ + ​​b​​ ​ = ​​c​​ ​

Try It! 1. Find the unknown side length of each right triangle. a. AB b. EF B F 7 15 E D 10 A C 12

LESSON 9-6 Right Triangles and the Pythagorean Theorem 453 Activity Assess

APPLICATION EXAMPLE 2 Use the Pythagorean Theorem and Its Converse

A. To satisfy safety regulations, the from the wall to the of a ladder should be at least one- fourth the length of the ladder. Did Drew set up the ladder correctly? The floor, the wall, and the ladder

form a right triangle. 9 ft

Step 1 Find the length of the ladder. 2 2 2 a ​​ ​ + ​b​​ ​ = ​c​​ ​ 2 2 2 ​​2 . 5 ​​ ​ + ​9​​ ​ = ​c​​ ​ 2.5 ft 2 87.25​ = ​c​​ ​ ​ Use the Pythagorean Theorem 9.34 ≈ c​ with ​a = 2.5 and b = 9​. 1 Step 2 Find __​​ ​​ the length of the ladder. 4 1 1 ​​ __ ​ c __​ ​(9.34)​ 4 ≈ 4 The length of the ladder is 9.34 ft. ≈ 2.335​ Since ​2.5 > 2.335​, Drew set up the ladder correctly.

B. The length of each crosspiece of the fence is 10 ft. Why would a rancher build this fence with the measurements shown? The numbers 6, 8, and 10 form a Pythagorean triple. A Pythagorean STUDY TIP triple is a set of three nonzero whole 6 ft Learn and recognize common numbers that satisfy the equation 2 2 2 Pythagorean triples such as 3, 4, ​​a​​ ​ + ​b​​ ​ = ​c​​ ​. 8 ft and 5; and 5, 12, and 13 to Since ​​6​​ 2​ ​8​​ 2​ 1​0 ​​ 2​​, the posts, the calculations. + = ground, and the crosspieces form right triangles. By using those measurements, the rancher knows that the fence posts are to the ground, which stabilizes the fence.

Try It! 2. a. What is KL? K

9 cm

J L 40 cm

b. Is △​ MNO​ a right triangle? N 37 cm Explain. O 35 cm 12 cm M

454 TOPIC 9 Similarity and Right Triangles Go Online | PearsonRealize.com Activity Assess CONCEPTUAL UNDERSTANDING EXAMPLE 3 Investigate Side Lengths in ​45°-45°-90°​ Triangles

Is there a relationship between the lengths of AB‾​​ ​​ A and ​​AC‾ ​​ in △​ ABC​ ? Explain. REASON Draw altitude CD‾​​ ​​ to form similar right triangles ​ D Think about the properties of △ABC​, △​ ACD​, and △​ CBD​. a triangle with two congruent . How do the properties of Notice that △​ ABC​ is a ​ 45 the triangle help you relate the C B 45 45 90 ​ triangle, side lengths? °- °- ° and that ​AC = BC​.

Use right-triangle similarity to write an equation. AB AC ​​ ___ ​ = ___​ ​ AC AD Since △​ ABC ∼ △ACD​, AC is the AB AC of AB and AD. ___​​ ​ ​ ____ ​ = 1 AC ​ __ ​AB 2 1 __​​ ​A​B​​ 2​ ​AC​​ 2​ 2 = Because ​△ABC​ is isosceles, ​​ 2 2 CD‾ ​​ bisects AB‾​​ ​​. AB​​ ​ = ​2AC​​ ​ __ AB = ​√ 2 ​ ∙ AC​ __ The length of AB‾​​ ​​ is √​​ 2 ​​ the length of AC‾​​ ​​.

Try It! 3. Find the side lengths of each 45°-45°-90° triangle.

a. What are XZ and YZ? b. What are JK and LK?

X L 45 7 K 45 45 Y Z 12√2 45

J

THEOREM 9-10 4 5 °-45°-90°​ Triangle Theorem

In a ​45°-45°-90°​ triangle, the legs If... B are congruent and the__ length of the hypotenuse is √​​ 2 ​​ times the 45 length of a leg. s

45 A s C __ PROOF: SEE EXERCISE 18. Then... BC = s√​​ 2 ​​

LESSON 9-6 Right Triangles and the Pythagorean Theorem 455 Activity Assess

EXAMPLE 4 Explore the Side Lengths of a ​30°-60°-90°​ Triangle Using an , show how the lengths of the short leg, the long leg, and the hypotenuse of a ​30°-60°-90°​ triangle are related. B ​△ABC is an equilateral ​ Altitude BD‾​​ ​​ divides △​ ABC​ into triangle. two congruent ​30°-60°-90°​ 30 30 triangles, ​△ADB​ and △​ CDB​.

60 60 A C D

Look at △​ ADB​ . Let the length of the short leg AD‾​​ ​​ be s. STUDY TIP Recall that an altitude of a Find the relationship between AD and AB. triangle is perpendicular to a side. AD = CD = s Think about what properties of ​​BD‾ ​​ bisects AC‾​ ​​. the triangle result in the altitude AC = AD + CD also being a segment bisector. AC = 2s ​△​ABC is equilateral, so AB = AC = 2s. AB = 2s Find the relationship between AD and BD. ​​AD​​ 2​ ​​BD​​ 2​ ​​AB​​ 2​ + = Use the Pythagorean Theorem. 2 2 2 s​​ ​ + ​​BD​​ ​ = ​​(2s)​​ ​ 2 2 BD​​ ​ = ​​3 s​​ ​ __ BD = s​​√ 3 ​​

In ​△ADB​ , the length of hypotenuse AB‾​​ __ ​​ is twice the length of the short leg ​​ AD‾ ​​. The length of the long leg BD‾​​ ​​ is √​​ 3 ​​ times the length of the short leg.

Try It! 4. a. What are PQ and PR? b. What are UV and TV?

R 9√3 T U 30 60 6 60 30 P Q V

THEOREM 9-11 3 0 °-60°-90°​ Triangle Theorem

In a ​30°-60°-90°​ triangle, the length If... A of the hypotenuse is twice the length of the short leg. The length __ 30 of the long leg is ​​√ 3 ​​ times the length of the short leg.

60 C s B __ PROOF: SEE EXERCISE 19. Then... ​AC = s√​ 3 ​​ , ​AB = 2s​

456 TOPIC 9 Similarity and Right Triangles Go Online | PearsonRealize.com Activity Assess

EXAMPLE 5 Apply Relationships

A. Alejandro needs to make both the horizontal B and vertical supports, AC‾​​ ​​ and AB‾​​ ​​, for the ramp. 60 Is one 12- board long enough for both 10 ft supports? Explain. COMMON ERROR 30 A C Be careful not to mix up the The ramp and supports form a ​30°-60°-90°​ relationship of the shorter and triangle. longer legs. Remember that the __ __ ​ BC 2AB AC AB​√3 ​​ √ = = longer leg is ​​ 3 ​​ times as long __ as the shorter leg, so the longer 1 0 = 2AB AC = 5​√3 ​​ ft 1 leg is between 1_​​ ​​ and 2 times as 2 ​AB 5​ ft long as the short leg. = Find the total length of the supports. __ ​AB + AC = 5 + 5√​​​ 3 ​​ ​≈ 13.66​ ft Since ​13.66 > 12​, the 12-foot board will not be long enough for Alejandro to make both supports.

B. Olivia starts an origami paper crane by making the 200-mm fold. What are the side length and area of the paper square? Step 1 Find the length of one side of the paper. __ s​​√ 2 ​​ = 200 200 s = ​​ ______​ ​√ 2 ​ fold s 141.4 mm ≈ s

Step 2 Find the area of the paper square. 45° 45° 2 A = ​​s​​ ​ 200 mm __ 2 A = (​​​ 100​√ 2 ​ )​ ​ 2 A = 20,000 m​​m​​ ​ The paper square has side length 141.4 mm and area 20,000 ​ mm​​ 2​.

Try It! 5. a. What are AB and BC? b. What are AC and BC?

B B 60 45 14 30 A C 28 45 A C

LESSON 9-6 Right Triangles and the Pythagorean Theorem 457 Concept Summary Assess CONCEPT SUMMARY The Pythagorean Theorem and Special Right Triangles

Converse of the THEOREM 9-8 Pythagorean Theorem THEOREM 9-9 Pythagorean Theorem

If... ​△ABC​ is a right triangle 2 2 2 If... ​​a​​ ​ + ​b​​ ​ = ​c​​ ​ B B c a c a A C b A C b 2 2 2 Then... ​​a​​ ​ + ​b​​ ​ = ​c​​ ​ Then... ​△ABC​ is a right triangle.

THEOREM 9-10 45°-45°-90° Triangle Theorem THEOREM 9-11 30°-60°-90° Triangle Theorem If... B If... A

45 s 30

45 A C 60 C B __ s Then... ​BC s√​ 2 ​​ = __ Then... ​AC = s√​ 3 ​, AB = 2s​

Do You UNDERSTAND? Do You KNOW HOW?

1. ESSENTIAL QUESTION How are For Exercises 4 and 5, find the value of x. similarity in right triangles and the 4. 6.4 5. x B C E F Pythagorean Theorem related? 45 x 2. Error Casey was asked to find XY. 60 18 What is Casey’s error? A D

For Exercises 6–8, is △​ RST​ a right triangle? XY = YZ √3 Y Explain. XY = 4√3 60° 4 6. ​RS = 20, ST = 21, RT = 29​ 30° X Z 7. ​RS = 35, ST = 36, RT = 71​ ✗ 8. ​RS = 40, ST = 41, RT = 11​ 9. Charles wants to hang the pennant shown vertically between two windows that 41 in. 30 are 19 apart. Will the 3. Reason A right triangle has leg lengths __ pennant fit? Explain. 4.5 and 4.5√​​ 3 ​​. What are the measures of the angles? Explain.

458 TOPIC 9 Similarity and Right Triangles Go Online | PearsonRealize.com Scan for Practice Tutorial PRACTICE & PROBLEM SOLVING Multimedia Additional Exercises Available Online

UNDERSTAND PRACTICE

10. Mathematical Connections Which rectangular For Exercises 15 and 16, find the unknown side has the longer diagonal? Explain. length of each triangle. SEE EXAMPLE 1 Prism P Prism Q 15. RS 16. XY T Y

5√10 10 15 9 9 X Z R S 13 3 4 4 12 17. Given △​ ABC​ with B 2 2 2 ​​a​​ ​ + ​​b​​ ​ = ​​c​​ ​​, write a 11. Error Analysis Dakota is 21 D E paragraph proof of c asked to find EF. What is a the Converse of the her error? Pythagorean Theorem. A C b F SEE EXAMPLE 2 There is not enough information 18. Write a two-column proof K to ﬁnd EF because you need to of the 45°-45°-90° Triangle know either the length of DF or Theorem. SEE EXAMPLE 3 one of the other measures. 45 45 J L

✗ 19. Write a paragraph proof A of the 30°-60°-90° Triangle Theorem. SEE EXAMPLE 4 30

12. Make Sense and Persevere What are expressions for MN and LN? Hint: Construct the altitude from M to LN‾​​ ​​. 60 C B M x For Exercise 20 and 21, find the side lengths of each triangle. SEE EXAMPLES 3 AND 4 45 30 L N 20. What are GJ and HJ? 21. What are XY and YZ? 13. Higher Order Thinking H Y Triangle XYZ is a right Y triangle. For what kind 12 45 60 30 of triangle would X Z 2 2 2 18 ​​XZ​​ ​ + ​XY​​ ​ > ​YZ​​ ​​? For what 45 G J kind of triangle would ​​ X Z 2 2 2 XZ​​ ​ + ​XY​​ ​ < ​YZ​​ ​​? Explain. 22. What is QS? SEE EXAMPLE 5 14. Look for Relationships Write an equation R that represents the relationship between JK and KL. L Q 30 45 S 10 T

30 K J

LESSON 9-6 Right Triangles and the Pythagorean Theorem 459 Practice Tutorial PRACTICE & PROBLEM SOLVING Mixed Review Available Online

APPLY ASSESSMENT PRACTICE

23. Reason Esteban wants marble bookends cut 26. Match each set of triangle side lengths with the at a 60° angle, as shown. If Esteban wants his best description of the triangle. ______bookends to be between 7.5 in. and 8 in. tall, I. √ 2 ​​ , ​​√ 2 ​​, ​​√ 3 ​​ A. right triangle what length d should the marble cutter make __ ___ the base of the bookends? Explain. II. 5, 3​​√ 2 ​​, ​​√ 4 3 ​​ B. 30°-60°-90° triangle __ III. 8, 8, 8√​​ 2 ​​ C. 45°-45°-90° triangle __ IV. 11, 11√​​ 3 ​​, 22 D. not a right triangle 27. SAT/ACT What is GJ?

H

60° 30 d 37.4

24. Communicate Precisely Sarah finds an antique 60 dinner bell that appears to be in the G J of an isosceles right triangle, but the only __ measurement given is the longest side. Ⓐ 18.7 Ⓒ 18.7​​√ 3 ​​ Sarah wants to display the bell and wand __ 18.7​​√ 2 ​​ 74.8 in a 5.5-in. by 7.5-in. picture frame. Assuming Ⓑ Ⓓ that the bell is an isosceles right triangle, 28. Performance Task Emma designed two can Sarah display the bell and wand within the triangular sails for a boat. frame? Explain.

45 30 12 m 5.5 in. 8.7 m 7 in.

45 60 1.5 in. 7.5 in. Part A What is the area of Sail A? 25. Construct Arguments When Carmen parks on a hill, she places chocks behind the wheels of Part B What is the area of Sail B? her car. The height of the chocks must be at least one-fourth of the height of the wheels to Part C Is it possible for Emma to cut both sails hold the car securely in place. The chock shown from one square of sailcloth with sides that are has the shape of a right triangle. Is it safe for 9 meters in length? Draw a diagram to explain. Carmen to use? Explain.

24 in.

9 in.

6 in.

460 TOPIC 9 Similarity and Right Triangles Go Online | PearsonRealize.com