8-2 Special Right 8-2 1. Plan

What You’ll Learn Check Skills You’ll Need GO for Help Lesson 1-6 Objectives 1 To use the properties of • To use the properties of 458-458-908 triangles Use a protractor to find the measures of the of each . 45°-45°-90° Triangles 2 To use the properties of • To use the properties of 1. 2. 3. 30°-60°-90° Triangles 308-608-908 triangles 2 2 4 2 32 3 . . . And Why Examples 1 Finding the Length of the To find the distance from 2 2 3 home plate to second base 45, 45, 90 30, 60, 90 3 on a softball diamond, as 45, 45, 90 2 Finding the Length of a Leg in Example 3 3 Real-World Connection 4 Using the Length of One Side 5 Real-World Connection

1 Using 458-458-908 Triangles 1 Math Background 8 The acute angles of an isosceles are both 45 angles. Another name The ratio of the lengths of any 8 8 8 for an isosceles right triangle is a 45 -45 -90 triangle. If each leg has length x and two sides of a right triangle is a the hypotenuse has length y, you can solve for y in terms of x. function of either acute . x2 + x2 = y2 Use the . This can be proved using similarity x x theorems and is the basis for 2x2 = y2 Simplify. 45Њ 45Њ the six . x 2 = y Take the root of each side. y ! The fixed side-length ratios of You have just proved the following theorem. 45°-45°-90° and 30°-60°-90° triangles, easily found by applying the Pythagorean Theorem, Key Concepts Theorem 8-5 458-458-908 Triangle Theorem provide benchmark values for the trigonometric functions , In a 458-458-908 triangle, both legs are congruent and the s2 45Њ cosine, and tangent of 30°, 45°, length of the hypotenuse is 2 times the length of a leg. s ! and 60° angles. Њ hypotenuse = 2 ? leg 45 D E ! s B C 1 A D E B C More Math Background: p. 414C 2 A D E B C 3 A D E B C 4 A D E B C 5 A D E B C Test-Taking Tip If you forget the 1 EXAMPLE Finding the Length of the Hypotenuse Lesson Planning and formula for a Resources 45°-45°-90° triangle, Find the value of each variable. use the Pythagorean See p. 414E for a list of the Theorem. The triangle a. b. 45Њ is isosceles, so the legs 9 resources that support this lesson. have the same length. Њ Њ x 2 2 45 45 45Њ h PowerPoint

h = 2 ? 9 hypotenuse ≠ 2 ? leg x = 2 ? 2 2 Bell Ringer Practice ! ! ! ! h = 9 2 Simplify. x = 4 ! Check Skills You’ll Need For intervention, direct students to: Quick Check 1 Find the length of the hypotenuse of a 458-458-908 triangle with legs of length 53 . ! Measuring Angles 5 6 " Lesson 1-6: Example 2 Lesson 8-2 Special Right Triangles 425 Extra Skills, Word Problems, Proof Practice, Ch. 1

Special Needs L1 Below Level L2 For Example 3, have students check the answer by In the diagram for Theorem 8-6, construct a 30° angle cutting out a 60-mm by 60-mm square. They fold it adjacent to the 30° angle, using a leg as one side. along its diagonal, and measure the length of the Extend the base so that it intersects the new side. Discuss diagonal to the nearest millimeter. why this forms an . learning style: tactile learning style: visual 425 2. Teach 2 EXAMPLE Finding the Length of a Leg Multiple Choice What is the value of x? 2 2 33 66 45Њ 6 Guided Instruction ! ! x 6 = 2 ? x hypotenuse ≠ 2 ? leg !6 ! x = Divide each side by 2 . 45Њ 1 EXAMPLE Technology Tip 2 ! ! = 6 ? 2 = 6 2 Point out that using mental x 2 !2 !2 Multiply by a form of 1. ! ! math is much faster than using a x = 3 2 Simplify. calculator for part b. The calculator ! The correct answer is B. answer also would be inexact, whereas squaring the square root Quick Check 2 Find the length of a leg of a 458-458-908 triangle with a hypotenuse of length 10. of a number is always exact. 5 2 " 2 EXAMPLE Auditory Learners When you apply the 458-458-908 Triangle Theorem to a real-life example, you can Have several students explain use a calculator to evaluate square roots. aloud to the class how to rationalize a denominator. 3 EXAMPLE Real-World Connection

PowerPoint Softball A high school softball diamond is a square. The distance from base to base is 60 ft. To the nearest foot, how far does a catcher throw the ball from home Additional Examples plate to second base? 1 Find the length of the The distance d from home plate to second 60 ft hypotenuse of a 45°-45°-90° base is the length of the hypotenuse of a triangle with legs of length 56 . 458-458-908 triangle. " Real-World Connection d 10 3 d = 60 2 hypotenuse ≠ 2 ? leg " Careers Opportunities for ! ! = 84.852814 2 Find the length of a leg coaching in women’s sports d Use a calculator. of a 45°-45°-90° triangle with have soared since the passage of Title IX in 1972. On a high school softball diamond, the catcher throws the ball about 85 ft from a hypotenuse of length 22. 11 2 " home plate to second base. 3 The distance from one corner Quick Check 3 A square garden has sides 100 ft long. You want to build a brick path along a to the opposite corner of a square diagonal of the square. How long will the path be? Round your answer to the playground is 96 ft. To the nearest nearest foot. 141 ft foot, how long is each side of the playground? 68 ft

Visual Learners 12 Using 308-608-908 Triangles Suggest that students distinguish between the 45°-45°-90° and the Another type of special right triangle is a 308-608-908 triangle. 30°-60°-90° Triangle Theorems by using the “ratio” diagrams below. Key Concepts Theorem 8-6 308-608-908 Triangle Theorem In a 308-608-908 triangle, the length of the hypotenuse ° 3 30 2 is twice the length of the shorter leg. The length of the 2 30Њ 1 45° longer leg is 3 times the length of the shorter leg. 2s ! s3 45° 60° hypotenuse = 2 ? shorter leg 1 1 60Њ longer leg = 3 ? shorter leg s Error Prevention! ! Whenever the length of a hypotenuse or longer leg of a 30°-60°-90° triangle is given, encourage students to find the 426 Chapter 8 Right Triangles and length of the shorter leg first.

Advanced Learners L4 English Language Learners ELL After students learn and apply Theorem 8-5, have Ask students to complete each statement: The them write a formula for the of an isosceles right shortest side of a triangle is always opposite the triangle whose hypotenuse has length s. smallest angle. In a 30°-60°-90° triangle, the shortest side is always opposite the 30° angle . 426 learning style: verbal learning style: verbal 4 EXAMPLE Math Tip To prove Theorem 8-6, draw a 308-608-908 triangle using an equilateral triangle. Students can use the Pythagorean Proof Proof of Theorem 8-6 W Theorem to check their work. For 308-608-908 #WXY in equilateral #WXZ, PowerPoint WY is the perpendicular bisector of XZ . Њ 1 1 30 Thus, XY = 2 XZ = 2 XW, or XW = 2XY = 2s. Additional Examples Also, 60Њ 60Њ 4 The longer leg of a 30°-60°-90° XY 2 + YW2 = XW2 Use the Pythagorean Theorem. X Z s Y triangle has length 18. Find the 2 + 2 = 2 s YW (2s) Substitute s for XY and 2s for XW. lengths of the shorter leg and the YW2 = 4s2 - s2 Subtract s2 from each side. hypotenuse. shorter leg: 63 ; " 2 = 2 hypotenuse: 12 3 YW 3s Simplify. " YW = s 3 Find the square root of each side. ! 5 A garden shaped like a rhombus has a perimeter of 100 ft and a 60° angle. Find the The 30°-60°-90° Triangle Theorem, like the 45°-45°-90° Triangle Theorem, lets you perpendicular height between two find two sides of a triangle when you know the length of the third side. bases. 21.7 ft

4 EXAMPLE Using the Length of One Side Resources • Daily Notetaking Guide 8-2 L3 Algebra Find the value of each variable. • Daily Notetaking Guide 8-2— d 5 5 = d 3 longer leg ≠ 3 ? shorter leg Adapted Instruction L1 ! Á 60Њ 30Њ 5 3 5 3 d = ? ! = ! Solve for d. f 3 3 3 " ! f = 2d hypotenuse ≠ 2 ? shorter leg Closure = ? 5 3 = 10 3 5 3 f 2 !3 3! Substitute !3 for d. In quadrilateral ABCD, AD = DC Њ Quick Check 4 Find the value of each variable. x = 4, y = 4 3 8 60 and AC = 20. Find the area of " x 30Њ ABCD. Leave your answer in y simplest radical form. C 5 EXAMPLE Real-World Connection 75° B Road Signs The moose warning sign at the left is an equilateral triangle. The height of the sign is 1 m. Find the length s of each side of the sign to the nearest tenth s 105° A D of a meter. 1 m 60Њ 100 ± 50 3 The triangle is equilateral, so the divides the " 1 triangle into two 30°-60°-90° triangles as shown in the 2 s diagram. The altitude also bisects the base, so the shorter 1 leg of each 30°-60°-90° triangle is 2 s. 1 1 = 3 s longer leg ≠ 3 ? shorter leg ! Q2 R Á 2 5 s 3 Solve for s. ! s ≈ 1.155 Simplify. Use a calculator. Each side of the sign is about 1.2 m long.

Quick Check 5 If the sides of the sign are 1 m long, what is the height? about 0.9 m

Lesson 8-2 Special Right Triangles 427

427 3. Practice EXERCISES For more exercises, see Extra Skill, Word Problem, and Proof Practice. Practice and Problem Solving Assignment Guide A Practice by Example Find the value of each variable. If your answer is not an integer, leave it in simplest 1 AB1-8, 21, 22, 26 radical form. Example 1 ≠ ≠ x 2; y 2 y ≠ 60 2 (page 425) 1.x ≠ 8; y ≠ 8 2 2." 3. " 2 AB 9-20, 23-25 for " GO 45Њ y y 45Њ C Challenge 27-28 Help 8 x y 60 45Њ Test Prep 29-32 x 2 45Њ Mixed Review 33-41 Examples 2 and 3 4.x ≠ 15; y ≠ 15 5. 6. 10 (page 426) 45Њ 8 4 2 "y Homework Quick Check y 152 " 5 To check students’ understanding 45Њ 45Њ of key skills and concepts, go over x x 5 Exercises 4, 12, 18, 24, 25. 7. Dinnerware Design You are designing dinnerware. What is the length of Exercise 7 Ask a volunteer to a side of the smallest square plate on which a 20-cm chopstick can fit along bring chopsticks to class and a diagonal without any overhang? Round your answer to the nearest tenth demonstrate how to use them. of a centimeter. 14.1 cm Exercises 17–19 Students should 8. Helicopters The four blades of a helicopter meet at right angles and are first find any side length that can Exercise 7 all the same length. The distance between the tips of two adjacent blades be derived using a given side. is 36 ft. How long is each blade? Round your answer to the nearest tenth. After the first length is found, 25.5 ft the other lengths often fall Examples 4 and 5 x 2 Algebra Find the value of each variable. If your answer is not an integer, leave it in into place. (page 427) simplest radical form.

Exercises 20–22 Each of these 9. 10. 11. 30Њ y exercises requires constructing 40 x x an altitude to form a rectangle. 60Њ 30Њ y 10 Њ 30 23 y 60Њ ≠ ≠ x x ≠ 20; y ≠ 20 3 x 3; y 3 " " x ≠ 5; y ≠ 5 3 " 12. 13. 14. x 60Њ 2 3 GPS Guided Problem Solving L3 12 y 9 3 x Њ Enrichment L4 60Њ 30Њ 60 y x y Reteaching L2 ≠ ≠ ≠ x ≠ 4; y ≠ 2 ≠ ≠ 17. a 7; b 14; c 7; x ≠ 24; y ≠ 12 3 x 9; y 18 Adapted Practice L1 d ≠ 7 3 15. Architecture An" escalator lifts people to " PracticeName Class Date L3 the second floor, 25 ft above the first floor. ≠ ≠ Similar Polygons 18. a 6; b 6;2 Practice 8-2 " The escalator rises at a 30° angle. Are the polygons similar? If they are, write a similarity statement, and give the similarity ratio. If they are not, explain. c ≠ 2;3 d ≠ 6 25 ft X Q R A X How far does a person travel from the 1.A 2. 3. 3 3 " 5 6 44 Y 5 Њ 5 5 Z 10 12 20— 20— 30 Y 7 Z 3 3 bottom to the top of the escalator? 43 ft ST3 19. a ≠ 103 ; b ≠ 5;3 B 3 C BC14 MN5 " " 4.T 5.B 16 A K 6. XY4 ≠ ≠ JK8 8 c 15; d 5 16. City Planning Jefferson Park sits on one square city block 300 ft on each side. S 60؇ KL6 120؇ 12 8 20 35 88 9944 ؇ C 8 ?؇ 120 Sidewalks join opposite corners. About how long is each diagonal sidewalk 60 U N NM6 M 8 L 8 R 21 M W 4 Z 424 ft 17–19. See above left. LMNO M HIJK. Complete the proportions and statements. J M N K 2 7. ЄM ? 8. ЄK ? 9. ЄN ? B Apply Your Skills x Algebra Find the value of each variable. Leave your answer in simplest radical form. MN = ? HK = HI IJ = HK 10. IJ JK 11. ? LM 12. MN ? L O I H Algebra The polygons are similar. Find the values of the variables. 17. 18. 19. 13.C 14. P Q G L M b a 10 5 ft 5 in. 7 2 b 3.3 ft 3 in. a 4 3 a b 60Њ AB6 ft E x F SR8 in. O x N 45Њ 30Њ 30Њ 15.G 16. J K P x Q x 6 m H Њ Њ F S 9 m R 60 45 M L c d c d Y 15 m 1.5 cm X Z 10 cm 4 cm c d WE

kWXZ MkDFG. Use the diagram to find the following. G 17. the similarity ratio of ᭝WXZ and ᭝DFG Z 428 Chapter 8 Right Triangles and Trigonometry 18. mЄZ 19. DG 20. GF 3 37؇ 21. mЄG 22. mЄD 23. WZ W D 46XF

428 20.6 21.4 22. 8 & GO nline 60Њ 4. Assess Reteach 32 b Homework Help 23 a a 6 Њ Њ PowerPoint Visit: PHSchool.com 45 45 a Web Code: aue-0802 b b Lesson Quiz a ≠ 3; b ≠ 7 a ≠ 14; b ≠ 6 2 a ≠ 4; b ≠ 4 " 23. Rika; Sandra marked 23. Error Analysis Sandra drew the triangle at Use kABC for Exercises 1–3. the shorter leg as the right. Rika said that the lengths couldn’t 5 A opposite the 608 angle. be correct. With which student do you agree? 53 ؇ ؇ Explain your answer. 30 60 24. Open-Ended Write a real-life problem that 10 you can solve using a 308-608-908 triangle with a 12 ft hypotenuse. Show your solution. See margin. C B 25. Farming A conveyor belt carries bales of hay from the ground to the barn loft 18 GPS 24 ft above the ground. The belt makes a 608 angle with the ground. 1. If m&A = 45, find AC and AB. a. How far does a bale of hay travel from one end of the conveyor belt to the AC ≠ 18; AB ≠ 18 2 other? Round your answer to the nearest foot. 28 ft " & = b. The conveyor belt moves at 100 ft/min. How long does it take for a bale of 2. If m A 30, find AC and AB. AC ≠ 183 ; AB ≠ 36 hay to go from the ground to the barn loft? 0.28 min " 3. If m&A = 60, find AC 26. House Repair After heavy winds damaged a farmhouse, and AB. AC ≠ 6;3 workers placed a 6-m brace against its side at a 458 angle. " AB ≠ 12 3 Then, at the same spot on the ground, they placed a second, 30Њ " 8 4. Find the side length of a longer brace to make a 30 angle with the side of the house. 6 m a. How long is the longer brace? Round your answer to the 45°-45°-90° triangle with a Њ 4-cm hypotenuse. 2 2 N nearest tenth of a meter. 8.5 m 45 " 2.8 cm Exercise 25 b. How much higher on the house does the longer brace reach than the shorter brace? 3.1 m 5. Two 12-mm sides of a triangle form a 120° angle. Find the C Challenge 27. in 3 Dimensions Find the length length of the third side. 12 3 N 20.8 mm d, in simplest radical form, of the diagonal " of a cube with sides of the given length. See left. 2 27a. 3 units " a. 1 unit b. 2 units c. s units d b. 23 units d 1 " 2 c. s 3 units 1 Alternative Assessment " 28. Constructions Construct a 30°-60°-90° triangle 12 given a segment that is Have students use compass and a. the shorter leg. straightedge to construct a large b. the hypotenuse. equilateral triangle with one c. the longer leg. See back of book. altitude. Then have them explain how the three sides of one of the right triangles are related. Test Prep Test Prep Multiple Choice 29. What is the length of a diagonal of a square with sides of length 4? D A. 2 B.2 C. 2 2 D. 4 2 ! ! ! Resources 30. The longer leg of a 30°-60°-90° triangle is 6. What is the length of the For additional practice with a hypotenuse? H variety of test item formats: A. 2 3 B. 3 2 C. 4 3 D. 12 ! ! ! • Standardized Test Prep, p. 465 31. The hypotenuse of a 30°-60°-90° triangle is 30. What is the length of one of • Test-Taking Strategies, p. 460 its legs? D • Test-Taking Strategies with A. 3 10 B. 10 3 C. 15 2 D. 15 Transparencies ! ! !

lesson quiz, PHSchool.com, Web Code: aua-0802 Lesson 8-2 Special Right Triangles 429

24. Answers may vary. Sample: A ramp up to a door is 12 ft long. It has an incline of 30°. How high off the ground is the door? sol.: 6 ft 429 Short Response 32. The vertex angle of an is 120°. The length of its [2] a. 43 cm, OR about base is 24 cm. ! Use this Checkpoint Quiz to check 6.9 in. a. Find the height of the triangle from the vertex angle to its base. b. 83 cm, OR about b. Find the length of each leg of the isosceles triangle. students’ understanding of the ! skills and concepts of Lessons 8-1 13.9 in. [1] one correct part through 8-2. Resources MixedMixed ReviewReview Grab & Go • Checkpoint Quiz 1 for Lesson 8-1 For Exercises 33 and 34, leave your answers in simplest radical form. GO Help 33. A right traingle has a 6-in. hypotenuse and a 5-in leg. Find the length of the other leg. 11 in. " 34. An isosceles triangle has 20-cm legs and a 16-cm base. Find the length of the altitude to the base. 4 21 cm " Lesson 6-4 Determine whether each quadrilateral must be a parallelogram. If not, provide a counterexample.

35. no 35. The diagonals are congruent and perpendicular to each other. See margin. 2 36. Two opposite angles are right angles and two opposite sides are 5 cm long. yes 1 5 37. One pair of sides is congruent and the other pair of sides is parallel. no; an isosceles trapezoid 4 Lesson 4-3 Can you conclude that #TRY O#ANG from the given conditions? If so, name the postulate or theorem that justifies your conclusion. no 38. yes; AAS Thm. 38. &A > &T, &Y > &G, TR > AN 39. &T > &A, &R > &N, &Y > &G 40. &R > &N, TR > AN, TY > AG no 41. &G > &Y, &N > &R, RY > NG yes; ASA Post.

Checkpoint Quiz 1 Lessons 8-1 through 8-2

x 2 Algebra Find the value of each variable. Leave your answer in simplest radical form. x = 10; 1. 2. 3. y = 10 2 x 15 y ! y x 12 12 9 30Њ 10 x x = 123 ; y = 24 4.9 5. 6. ! y a 23 y x 30Њ 45Њ x 30Њ 9 2 x = 43 ; y = 6 15 ! x = 53 ; y = 10 3 The lengths of the sides of a triangle are given. Classify each triangle! as acute!, obtuse, or right. 7. 7, 8, 9 acute 8. 15, 36, 39 right 9. 10, 12, 16 obtuse

10. A square has a 40-cm diagonal. How long is each side of the square? Round your answer to the nearest tenth of a centimeter. 28.3 cm

430 Chapter 8 Right Triangles and Trigonometry

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