18.3 Special Right Triangles
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1 Name lass Date 18.3 Special Right Triangles Essential Question: What do you know about the side lengths and the trigonometric ratios in special right triangles? Eplore 1 Investigating an Isosceles Right Triangle Resource Locker Discover relationships that always apply in an isosceles right triangle. The figure shows an isosceles right triangle. Identify the base angles, and use the fact that they are complementary to write an equation relating their measures. Use the Isosceles Triangle Theorem to write a different equation relating the base angle measures. What must the measures of the base angles be? Why? D Use the Pythagorean Theorem to find the length of the hypotenuse in terms of the length of each leg,. Houghton Mifflin Harcourt Publishing ompany Reflect 1. Is it true that if you know one side length of an isosceles right triangle, then you know all the side lengths? Eplain. 2. What if? Suppose you draw the perpendicular from to. Eplain how to find the length of D. D 1 1 Module Lesson 3
2 Eplore 2 Investigating nother Special Right Triangle Discover relationships that always apply in a right triangle formed as half of an equilateral triangle. D is an equilateral triangle and is a perpendicular from to D. Determine all three angle measures in. Eplain why D. D Let the length of be. What is the length of, and why? D Using the Pythagorean Theorem, find the length of. Reflect 3. What is the numerical ratio of the side lengths in a right triangle with acute angles that measure and? Eplain. 4. Eplain the Error student has drawn a right triangle with a angle and a hypotenuse of 6. He has labeled the other side lengths as shown. Eplain how you can tell at a glance that he has made an error and how to correct it. L 3 J K Houghton Mifflin Harcourt Publishing ompany Module Lesson 3
3 Eplain 1 pplying Relationships in Special Right Triangles The right triangles you eplored are sometimes called --90 and --90 triangles. In a --90 triangle, the hypotenuse is 2 times as long as each leg. In a --90 triangle, the hypotenuse is twice as long as the shorter leg and the longer leg is 3 times as long as the shorter leg. You can use these relationships to find side lengths in these special types of right triangles Eample 1 Find the unknown side lengths in each right triangle. Find the unknown side lengths in. The hypotenuse is 2 times as long as each leg. = 2 = 2 Substitute 10 for. 10 = 2 = 2 Multiply by = 2 = 2 Divide by = = 10 In right DEF, m D = and m E =. The shorter leg measures 5 3. Find the remaining side lengths. Houghton Mifflin Harcourt Publishing ompany D E 5 3 The hypotenuse is twice as long as the shorter leg. DE = 2 Substitute for. DE = 2 Simplify. DE = The longer leg is 3 times as long as the shorter leg. = 3 Substitute for. = 3 F Simplify. = Module Lesson 3
4 Your Turn Find the unknown side lengths in each right triangle. 5. L 6. Q 4 3 J K P 2 6 R Eplain 2 Trigonometric Ratios of Special Right Triangles You can use the relationships you found in special right triangles to find trigonometric ratios for the angles,, and. Eample 2 For each triangle, find the unknown side lengths and trigonometric ratios for the angles triangle with a leg length of 1 Step 1 Since the lengths of the sides opposite the angles are congruent, they are both 1. The length of the hypotenuse is 2 times as long as each leg, so it is 1 ( 2 ), or 2. Step Use the triangle to find the trigonometric ratios for. Write each ratio as a simplified fraction. opp ngle Sine = hyp adj osine = hyp 2 2 opp Tangent = adj 1 Houghton Mifflin Harcourt Publishing ompany Module Lesson 3
5 --90 triangle with a shorter leg of 1 Step 1 The hypotenuse is as long as the leg, so the length of the hypotenuse is. The longer leg is times as long as the leg, so the length of the longer leg is. Step 2 1 Use the triangle to complete the table. Write each ratio as a simplified fraction. opp ngle Sine = hyp osine = adj hyp Tangent = opp adj Reflect 7. Write any patterns or relationships you see in the tables in Part and Part as equations. Why do these patterns or relationships make sense? 8. For which acute angle measure θ, is tanθ less than 1? equal to 1? greater than 1? Houghton Mifflin Harcourt Publishing ompany Your Turn Find the unknown side lengths and trigonometric ratios for the angles Module Lesson 3
6 Eplain 3 Investigating Pythagorean Triples Pythagorean Triples Pythagorean triple is a set of positive integers a, b, and c that satisfy the equation a 2 + b 2 = c 2. This means that a, b, and c are the legs and hypotenuse of a right triangle. Right triangles that have non-integer sides will not form Pythagorean triples. Eamples of Pythagorean triples include 3, 4, and 5; 5, 12, and 13; 7, 24, and 25; and 8, 15, and 17. c b a Eample 3 Use Pythagorean triples to find side lengths in right triangles. Verify that the side lengths 3, 4, and 5; 5, 12, and 13; 7, 24, and 25; and 8, 15, and 17 are Pythagorean triples. The numbers in Step are not the only Pythagorean triples. In the following steps you will discover that multiples of known Pythagorean triples are also Pythagorean triples. In right triangles DEF and JKL, a, b, and c form a Pythagorean triple, and k is a positive integer greater than 1. Eplain how the two triangles are related. J D a F b c E ka L kc kb K DEF is similar to by the because the corresponding sides are proportional. omplete the ratios to verify Side-Side-Side (SSS) Triangle Similarity. a : : c = : : You can use the Pythagorean Theorem to compare the lengths of the sides of JKL. What must be true of the set of numbers ka, kb, and kc? (ka 2 ) + (kb 2 ) = + k 2 b 2 = k 2 Houghton Mifflin Harcourt Publishing ompany = k 2 ( c 2 ) = The set of numbers ka, kb, and kc form a. Module Lesson 3
7 Reflect 10. Suppose you are given a right triangle with two side lengths. What would have to be true for you to use a Pythagorean triple to find the remaining side length? Your Turn Use Pythagorean triples to find the unknown side length. 11. R In XYZ, the hypotenuse XY has length 68, and the shorter leg XZ has length 32. P 18 Q Elaborate 13. Describe the type of problems involving special right triangles you can solve. Houghton Mifflin Harcourt Publishing ompany 14. How can you use Pythagorean triples to solve right triangles? 15. Discussion How many Pythagorean triples are there? 16. Essential Question heck-in What is the ratio of the length of the hypotenuse to the length of the shorter leg in any --90 triangle? Module Lesson 3
8 Evaluate: Homework and Practice For each triangle, state whether the side lengths shown are possible. Eplain why or why not. Online Homework Hints and Help Etra Practice Find the unknown side lengths in each right triangle L 9 J K Houghton Mifflin Harcourt Publishing ompany Module Lesson 3
9 7. Right triangle UVW has acute angles U measuring and W measuring. Hypotenuse UW measures 12. (You may want to draw the triangle in your answer.) 8. Right triangle PQR has acute angles P and Q measuring. Leg PR measures (You may want to draw the triangle in your answer.) Use trigonometric ratios to solve each right triangle D 14 E 10 3 F Houghton Mifflin Harcourt Publishing ompany 11. Right KLM with m J =, leg JK = Right PQR with m Q =, leg QR = 15 J 4 3 L K P Q 15 R Module Lesson 3
10 For each right triangle, find the unknown side length using a Pythagorean triple. If it is not possible, state why K J 4 L 15. In right PQR, the legs have lengths PQ = 9 and QR = In right XYZ, the hypotenuse XY has length 35, and the shorter leg YZ has length Solve for. 12 E 12 D H F G 18. Represent Real-World Problems baseball diamond actually forms a square, each side measuring 30 yards. How far, to the nearest yard, must the third baseman throw the ball to reach first base? Houghton Mifflin Harcourt Publishing ompany Image redits: Eactostock/Superstock Module Lesson 3
11 19. In a right triangle, the longer leg is eactly 3 times the length of the shorter leg. Use the inverse tangent trigonometric ratio to prove that the acute angles of the triangle measure and. lgebra Find the value of in each right triangle L J (3-25) 2 K Houghton Mifflin Harcourt Publishing ompany 22. Eplain the Error harlene is trying to find the unknown sides of a right triangle with a acute angle, whose hypotenuse measures Identify, eplain, and correct harlene s error. P Q 12 R Module Lesson 3
12 23. Represent Real-World Problems Honeycomb blinds form a string of almostregular heagons when viewed end-on. pproimately how much material, to the nearest ten square centimeters, is needed for each 3.2-cm deep cell of a honeycomb blind that is 125 cm wide? (Hint: Draw a picture. regular heagon can be divided into 6 equilateral triangles.) cm cm 24. Which of these pairs of numbers are two out of three integer-valued side lengths of a right triangle? (Hint: for positive integers a, b, c, and k, ka, kb, and kc are side lengths of a right triangle if and only if a, b, and c are side lengths of a right triangle.). 15, 18 True False. 15, 30 True False. 15, 51 True False D. 16, 20 True False E. 16, 24 True False Houghton Mifflin Harcourt Publishing ompany Image redits: Eactostock/Superstock Module Lesson 3
13 H.O.T. Focus on Higher Order Thinking 25. ommunicate Mathematical Ideas Is it possible for the three side lengths of a right triangle to be odd integers? Eplain. 26. Make a onjecture Use spreadsheet software to investigate this question: are there sets of positive integers a, b, and c such that a 3 + b 3 = c 3? You may choose to begin with these formulas: Houghton Mifflin Harcourt Publishing ompany Module Lesson 3
14 Lesson Performance Task Kate and her dog are longtime flying disc players. Kate has decided to start a small business making circles of soft material that dogs can catch without injuring their teeth. Since she also likes math, she s decided to see whether she can apply Pythagorean principles to her designs. She used the Pythagorean triple for the dimensions of her first three designs. r = 3 in. r = 4 in. r = 5 in. small medium large 1. Is it true that the (small area) + (medium area) = (large area)? Eplain. 2. If the circles had radii based on the Pythagorean triple , would the above equation be true? Eplain. 3. Three of Kate s circles have radii of a, b, and c, where a, b, and c form a Pythagorean triple ( a 2 + b 2 = c 2 ). Show that the sum of the areas of the small and medium circles equals the area of the large circle. 4. Kate has decided to go into the beach ball business. Sticking to her Pythagorean principles, she starts with three spherical beach balls--a small ball with radius 3 in., a medium ball with radius 4 in., and a large ball with radius 5 in. Is it true that (small volume) + (medium volume) = (large volume)? Show your work. 5. Eplain the discrepancy between your results in Eercises 3 and 4. Houghton Mifflin Harcourt Publishing ompany Image redits: dlewis33/istockphoto.com Module Lesson 3
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