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Page 1 of 6 10.2 Goal Find the side lengths of 458-458-908 triangles. Key Words 458-458-908 Triangles Geo-Activity Exploring an Isosceles Right Triangle 1 Fold a large piece of paper so the top lines up with one side. ● 2 Measure the ● • 458-458-908 triangle • isosceles triangle p. 173 3 Measure the ● angles of the triangle formed. legs of the triangle. • leg of a right triangle p. 192 • hypotenuse p. 192 4 Use the Pythagorean Theorem to predict the length of the hypotenuse. ● 5 Measure the hypotenuse to verify your answer in Step 4. ● Student Help LOOK BACK To review the Pythagorean Theorem, see p. 192. A right triangle with angle measures of 458, 458, and 908 is called a 458-458-908 triangle . You can use the Pythagorean Theorem to find the length of the hypotenuse of any 458-458-908 triangle. 2 2Ï· 2 2 3Ï· 3 2 3 Ï2w2w1 ww22w 5 Ï4ww 1w 4 Ï3w2w1 ww 3 2w 5 Ï9 ww 1w 9 2 4Ï· 4 4 Ï4w2w1 ww42w 5 Ï1w6ww 1w6 1w 5 Ï4 wwpw 2 5 Ï9 wwpw 2 5 Ï1 w6 wwpw 2 5 Ï4 w p Ï2 w 5 Ï9 w p Ï2 w 5 Ï1 w6 w p Ï2 w 5 2Ï2 w 5 3Ï2 w 5 4Ï2 w THEOREM 10.1 458-458-908 Triangle Theorem Words In a 458-458-908 triangle, the length of the hypotenuse is the length of a leg times Ï2 w. Symbols hypotenuse 5 leg p Ï2w 542 Chapter 10 Right Triangles and Trigonometry x 458 x Ï2 458 x Page 2 of 6 IStudent Help EXAMPLE 1 Find Hypotenuse Length ICLASSZONE.COM MORE EXAMPLES More examples at classzone.com Find the length x of the hypotenuse in the 458-458-908 triangle shown at the right. 3 3 458 x Solution By the 458-458-908 Triangle Theorem, the length of the hypotenuse is the length of a leg times Ï2 w. w hypotenuse 5 leg p Ï2 5 3 p Ï2 w ANSWER 458-458-908 Triangle Theorem Substitute. © The length of the hypotenuse is 3Ï2 w. EXAMPLE 2 Find Leg Length Find the length x of each leg in the 458-458-908 triangle shown at the right. 7Ï· 2 Solution READING TIP By the 458-458-908 Triangle Theorem, the length of the hypotenuse is the length of a leg times Ï2 w. x hypotenuse 5 leg p Ï2 w 458-458-908 Triangle Theorem 7Ï2 w 5 xÏ2 w Substitute. 7Ï2 w w x Ï2 }5} Ï2 w Ï2 w Divide each side by Ï2 w. 75x ANSWER x 458 Student Help The expression x Ï2w is equivalent to Ï2wx . 458 Simplify. © The length of each leg is 7. Find Hypotenuse and Leg Lengths Find the value of x. 1. 2. x 458 5 458 3. x 458 458 4 4. x 458 458 6Ï· 2 458 458 x 3Ï· 2 10.2 458-45 8-90 8 Triangles 543 Page 3 of 6 EXAMPLE 3 Identify 458-458-908 Triangles Determine whether there is enough information to conclude that the triangle is a 458-458-908 triangle. Explain your reasoning. x8 x8 Solution By the Triangle Sum Theorem, x81 x8 1 908 5 1808. So, 2x8 5 908, and x 5 45. ANSWER © Since the measure of each acute angle is 458, the triangle is a 458-458-908 triangle. Example 3 shows that whenever a right triangle has congruent acute angles, it is a 458-458-908 triangle. Student Help EXAMPLE LOOK BACK 4 Find Leg Length Show that the triangle is a 458-458-908 triangle. Then find the value of x. To review the Base Angles Theorem, see p. 185. 5 x x Solution The triangle is an isosceles right triangle. By the Base Angles Theorem, its acute angles are congruent. From the result of Example 3, this triangle must be a 458-458-908 triangle. You can use the 458-458-908 Triangle Theorem to find the value of x. hypotenuse 5 leg p Ï2 w 5 5 xÏ2 w 458-458-908 Triangle Theorem Substitute. 5 xÏ2 w }5} Ï2 w Ï2 w Divide each side by Ï2 w. 5 }5x Ï2 w Simplify. 3.5 ≈ x Use a calculator to approximate. Find Leg Lengths Show that the triangle is a 458-458-908 triangle. Then find the value of x. Round your answer to the nearest tenth. 5. 6. x 12 x x 8 544 Chapter 10 Right Triangles and Trigonometry Page 4 of 6 10.2 Exercises Guided Practice Vocabulary Check 1. How many congruent sides does an isosceles right triangle have? 2. How many congruent angles does an isosceles right triangle have? What are the measures of the three angles? Skill Check Find the value of x in the 458-458-908 triangle. Write your answer in radical form. 3. 4. x 6 x 5. 2 2Ï· x Ï· 6 Ï·6 458 x Practice and Applications Extra Practice See p. 693. Finding Hypotenuse Lengths Find the length of the hypotenuse in the 458-458-908 triangle. Write your answer in radical form. 6. 7. 2 x 458 8. x 458 458 7 7 8 458 2 9. 10. 458 x 11. Ï· 3 458 Ï· 5 Ï· 3 458 x 458 Ï· 5 8 458 458 x Ï· 10· 458 x 458 Ï·10· Finding Leg Lengths Find the length of a leg in the 458-458-908 triangle. 12. 13. x 14. 10Ï· 2 4Ï· 2 x x x 2 5Ï· x x Homework Help 15. Example 1: Example 2: Example 3: Example 4: Exs. 6–11, 18 Exs. 12–17 Exs. 19–27 Exs. 22–27 x x Ï· 2 16. 17. 2 14Ï· x 8Ï· 2 x 10.2 x 458 458-45 8-90 8 Triangles 458 x 545 Page 5 of 6 18. Jewelry Use a calculator to find the length x of the earring shown at the right. Round your answer to the nearest tenth. 1.4 cm x 1.4 cm You be the Judge Determine whether there is enough information to conclude that the triangle is a 458-458-908 triangle. Explain your reasoning. 19. 20. 21. 458 Finding Leg Lengths Show that the triangle is a 458-458-908 triangle. Then find the value of each variable. Round to the nearest tenth. 22. 23. 4 x 24. 32 x 9 458 x x 25. 26. 8 x8 458 x8 y 27. 20 x x 35 458 Technology In Exercises 28–30, use geometry software. &* and construct its midpoint, C. 1 Draw AB ● &*. 2 Construct the perpendicular bisector of AB ● 3 Construct point D on the bisector and ● D &* and DB &*. construct AD 4 Measure aADB. Drag point D until ● A C maADB 5 908. 28. Name three 458-458-908 triangles. Explain how you know they are 458-458-908 triangles. &*, CB &*, and CD &*. What do you notice? Explain. 29. Measure AC &* and DB &*. Check your answer by 30. Predict the measures of AD measuring the segments. 31. Error Analysis A student labels a 458-458-908 triangle as shown. Explain and correct the error. Ï· 5 5Ï·2 Ï· 5 546 Chapter 10 Right Triangles and Trigonometry B Page 6 of 6 Quilt Design The quilt design in the photo is based on the pattern in the diagram below. Use the diagram in Exercises 32 and 33. 1 1 1 1 s t u r 1 1 v w 1 “Wheel of Theodorus,” by Diana Venters 32. Working from left to right, use the Pythagorean Theorem in each right triangle to find the values of r, s, t, u, v, and w. 33. Identify any 458-458-908 triangles in the figure. Standardized Test Practice 34. Multi-Step Problem Use the triangle shown below. a. Find the value of x. Then find A maA, maB, and maC. b. Find the values of b and c. b 3x8 c c. Use the Pythagorean Theorem or the 458-458-908 Triangle Theorem to justify your answers in part (b). Mixed Review 6x8 C 3x8 12 B Classifying Triangles Classify the triangle as acute, right, or obtuse. (Lesson 4.5) 35. 26 36. 10 9 24 40 37. 16 41 9 14 Simplifying Radicals Simplify the radical expression. (Lesson 10.1) Algebra Skills 38. Ï2 w4 w 39. Ï6 w3 w 40. Ï5 w2 w 41. Ï6 w4 w 42. Ï8 w0 w 43. Ï1 w9 w6 w 44. Ï2 w5 w0 w 45. Ï1 w1 w7 w Writing Fractions as Decimals Write the fraction as a decimal. For repeating decimals, also round to the nearest hundredth. (Skills Review, p. 657) 3 5 48. }} 3 20 52. }} 9 10 47. }} 4 9 51. }} 46. }} 50. }} 2 3 49. }} 47 50 53. }} 10.2 33 100 1 6 458-45 8-90 8 Triangles 547