8-3 Special Right Triangles Find x. 1. 3. SOLUTION: SOLUTION: In a 45°-45°-90° triangle, the legs are congruent ( In a 45°-45°-90° triangle, the legs are congruent ( = ) and the length of the hypotenuse h is times = ) and the length of the hypotenuse h is times the length of a leg . the length of a leg . Therefore, since the side length ( ) is 5, then Therefore, since , then . ANSWER: Simplify: ANSWER: 2. 22 SOLUTION: In a 45°-45°-90° triangle, the legs are congruent ( = ) and the length of the hypotenuse h is times the length of a leg . Therefore, since the hypotenuse (h) is 14, then Solve for x. ANSWER: eSolutions Manual - Powered by Cognero Page 1 8-3 Special Right Triangles Find x and y. 4. 5. SOLUTION: In a 30°-60°-90° triangle, the length of the SOLUTION: hypotenuse h is 2 times the length of the shorter leg s In a 30°-60°-90° triangle, the length of the (2s) and the length of the longer leg is times hypotenuse h is 2 times the length of the shorter leg s the length of the shorter leg ( ). (2s) and the length of the longer leg is times the length of the shorter leg ( ). The length of the hypotenuse is the shorter leg is y, and the longer leg is x. Therefore, The length of the hypotenuse is x, the shorter leg is 7, and the longer leg is y. Therefore, Solve for y: ANSWER: ; Substitute and solve for x: ANSWER: ; eSolutions Manual - Powered by Cognero Page 2 8-3 Special Right Triangles the triangle into two smaller congruent 30-60-90 triangles. Let x represent the length of the altitude and use the 30-60-90 Triangle Theorem to determine 6. the value of x. SOLUTION: In a 30°-60°-90° triangle, the length of the hypotenuse h is 2 times the length of the shorter leg s (2s) and the length of the longer leg is times the length of the shorter leg . The length of the hypotenuse is y, the shorter leg is x, The hypotenuse is twice the length of the shorter leg and the longer leg is 12. Therefore, s. Solve for x. Since the altitude is across from the 60º-angle, it is the longer leg. Substitute and solve for y: Since 3.25 < 3.5, the height of the plaque is less than ANSWER: the altitude of the equilateral triangle. Therefore, the plaque will fit through the opening of the mailer. ; ANSWER: 7. ART Paulo is mailing an engraved plaque that is Yes; sample answer: The height of the triangle is inches high to the winner of a chess tournament. He about in., so since the height of the plaque is less has a mailer that is a triangular prism with 4-inch than the height of the opening, it will fit. equilateral triangle bases as shown in the diagram. Will the plaque fit through the opening of the mailer? Explain. SOLUTION: If the plaque will fit, then the height of the plaque must be less than the altitude of the equilateral triangle. Draw the altitude of the equilateral triangle. Since the triangle is equilateral, the altitude will divide eSolutions Manual - Powered by Cognero Page 3 8-3 Special Right Triangles SENSE-MAKING Find x. 9. 8. SOLUTION: SOLUTION: In a 45°-45°-90° triangle, the legs are congruent ( In a 45°-45°-90° triangle, the legs are congruent ( = ) and the length of the hypotenuse h is times = ) and the length of the hypotenuse h is times the length of a leg . the length of a leg . Since the hypotenuse is 15 and the legs are x, then Therefore, since the hypotenuse is 16 and the legs are x, then Solve for x. Solve for x. ANSWER: ANSWER: or eSolutions Manual - Powered by Cognero Page 4 8-3 Special Right Triangles 10. 11. SOLUTION: SOLUTION: In a 45°-45°-90° triangle, the legs l are congruent In a 45°-45°-90° triangle, the legs are congruent and the length of the hypotenuse h is and the length of the hypotenuse h is times the length of a leg . times the length of a leg . Since the the legs are , then the hypotenuse is Therefore, since the legs are , the hypotenuse is . Simplify: Simplify: ANSWER: ANSWER: 34 12. SOLUTION: In a 45°-45°-90° triangle, the legs are congruent and the length of the hypotenuse h is times the length of a leg . Therefore, since the legs are 19.5, then the hypotenuse would be ANSWER: eSolutions Manual - Powered by Cognero Page 5 8-3 Special Right Triangles 15. Determine the length of the leg of a - - triangle with a hypotenuse length of 11. SOLUTION: In a 45°-45°-90° triangle, the legs are congruent 13. and the length of the hypotenuse h is SOLUTION: times the length of a leg . In a 45°-45°-90° triangle, the legs are congruent and the length of the hypotenuse h is Therefore, since the hypotenuse is 11, times the length of a leg . Solve for . Therefore, since the legs are 20, then the hypotenuse would be ANSWER: 14. If a - - triangle has a hypotenuse length of 9, find the leg length. SOLUTION: ANSWER: In a 45°-45°-90° triangle, the legs are congruent and the length of the hypotenuse h is times the length of a leg . 16. What is the length of the hypotenuse of a - - triangle if the leg length is 6 centimeters? Therefore, since the hypotenuse is 9, then SOLUTION: Solve for x. In a 45°-45°-90° triangle, the legs are congruent and the length of the hypotenuse h is times the length of a leg . Therefore, since the legs are 6, the hypotenuse would be ANSWER: or 8.5 cm ANSWER: eSolutions Manual - Powered by Cognero Page 6 8-3 Special Right Triangles 17. Find the length of the hypotenuse of a - - triangle with a leg length of 8 centimeters. SOLUTION: 19. In a 45°-45°-90° triangle, the legs are congruent SOLUTION: and the length of the hypotenuse h is In a 30°-60°-90° triangle, the length of the times the length of a leg . hypotenuse h is 2 times the length of the shorter leg s (h = 2s) and the length of the longer leg is Therefore, since the legs are 8, the hypotenuse would times the length of the shorter leg . be ANSWER: The length of the hypotenuse is 7, the shorter leg is x, and the longer leg is . or 11.3 cm Therefore, . Find x and y. Solve for x: 18. SOLUTION: Then , to find the hypotenuse, In a 30°-60°-90° triangle, the length of the ANSWER: hypotenuse h is 2 times the length of the shorter leg s x = 10; y = 20 (h=2s) and the length of the longer leg is times the length of the shorter leg . The length of the hypotenuse is y, the shorter leg is x, and the longer leg is . Therefore, . Solve for x: Then , to find the hypotenuse, ANSWER: x = 8; y = 16 eSolutions Manual - Powered by Cognero Page 7 8-3 Special Right Triangles 21. 20. SOLUTION: SOLUTION: In a 30°-60°-90° triangle, the length of the In a 30°-60°-90° triangle, the length of the hypotenuse h is 2 times the length of the shorter leg s hypotenuse h is 2 times the length of the shorter leg s (h = 2s) and the length of the longer leg is (h = 2s) and the length of the longer leg is times the length of the shorter leg . times the length of the shorter leg . The length of the hypotenuse is 17, the shorter leg is The length of the hypotenuse is 15, the shorter leg is y, and the longer leg is x. y, and the longer leg is x. Therefore, . Therefore, . Solve for y: Solve for y: Substitute and solve for x: Substitute and solve for x: ANSWER: ANSWER: ; ; eSolutions Manual - Powered by Cognero Page 8 8-3 Special Right Triangles 22. 23. SOLUTION: SOLUTION: In a 30°-60°-90° triangle, the length of the hypotenuse h is 2 times the length of the shorter leg s In a 30°-60°-90° triangle, the length of the hypotenuse h is 2 times the length of the shorter leg s (h = 2s) and the length of the longer leg is (h = 2s) and the length of the longer leg is times the length of the shorter leg . times the length of the shorter leg . The length of the hypotenuse is y, the shorter leg is 24, and the longer leg is x. The length of the hypotenuse is y, the shorter leg is Therefore, x, and the longer leg is 14. Therefore, . ANSWER: ; y = 48 Solve for x: Substitute and solve for y: ANSWER: ; eSolutions Manual - Powered by Cognero Page 9 8-3 Special Right Triangles 24. An equilateral triangle has an altitude length of 18 25. Find the length of the side of an equilateral triangle feet. Determine the length of a side of the triangle. that has an altitude length of 24 feet. SOLUTION: SOLUTION: Let x be the length of each side of the equilateral Let x be the length of each side of the equilateral triangle. The altitude from one vertex to the opposite triangle. The altitude from one vertex to the opposite side divides the equilateral triangle into two side divides the equilateral triangle into two 30°-60°-90° triangles. 30°-60°-90° triangles.