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Right Triangles and the Pythagorean Theorem Related? Activity Assess 9-6 EXPLORE & REASON Right Triangles and Consider △​ ABC​ with altitude ​​CD‾ ​​ as shown. the Pythagorean B Theorem D PearsonRealize.com A 45 C 5√2 I CAN… prove the Pythagorean Theorem using A. What is the area of △​ ​ABC? ​Of △​ACD? Explain your answers. similarity and establish the relationships in special right B. Find the lengths of ​​AD‾ ​​ and ​​AB‾ ​​. triangles. C. Look for Relationships Divide the length of the hypotenuse 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 • Pythagorean triple 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 triangle is a right triangle, If... ​△ABC​ is a right triangle. then the sum of the squares of the B lengths of the legs is equal to the square 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 equation 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 distance from the wall to the base 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 speed 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 perpendicular 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 angles. 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 geometric mean 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 ​​ times 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 equilateral triangle, 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 Special Right Triangle Relationships A. Alejandro needs to make both the horizontal B and vertical supports, ​​AC‾ ​​ and ​​AB‾ ​​, for the ramp. 60 Is one 12-foot 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 diagonal 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.
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