8-1 Geometric Mean Find the geometric mean between each pair of 4. Write a similarity statement identifying the three numbers. similar triangles in the figure. 1. 5 and 20 SOLUTION: By the definition, the geometric mean x of any two numbers a and b is given by Therefore, the geometric mean of 5 and 20 is SOLUTION: ANSWER: 10 2. 36 and 4 SOLUTION: By the definition, the geometric mean x of any two numbers a and b is given by Therefore, the geometric mean of 36 and 4 is If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. is the ANSWER: altitude to the hypotenuse of the right triangle CED. 12 Therefore, 3. 40 and 15 ANSWER: SOLUTION: By the definition, the geometric mean x of any two Find x, y, and z. numbers a and b is given by Therefore, the geometric mean of 40 and 15 is 5. ANSWER: SOLUTION: or 24.5 By the Geometric Mean (Altitude) Theorem the altitude drawn to the hypotenuse of a right triangle eSolutions Manual - Powered by Cognero Page 1 8-1 Geometric Mean separates the hypotenuse into two segments. The x = 6; ; . length of this altitude is the geometric mean between the lengths of these two segments. Solve for x. 6. SOLUTION: By the Geometric Mean (Leg) Theorem the altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of a leg of this triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Solve for y. By the Geometric Mean (Altitude) Theorem the altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of this altitude is the geometric mean between the lengths of these two segments. Solve for x. Solve for z. By the Geometric Mean (Leg) Theorem the altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of a leg of this triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg. ANSWER: Solve for y. eSolutions Manual - Powered by Cognero Page 2 8-1 Geometric Mean By the Geometric Mean (Leg) Theorem the altitude drawn to the hypotenuse of a right triangle separates SOLUTION: the hypotenuse into two segments. The length of a We have the diagram as shown, with feet converted leg of this triangle is the geometric mean between the to inches. length of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Solve for z. By the Geometric Mean (Altitude) Theorem the altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of this altitude is the geometric mean between ANSWER: the lengths of these two segments. x = 32; ; 7. MODELING Corey wants to estimate the height of the statue of Stephen F. Austin in Angleton, Texas. He stands approximately 19.3 feet away from the statue so that his line of vision to the top and base of the statue form a right angle, as shown in the diagram. About how tall is the statue? So, the total height of the statue is about 894 + 60 or 954 inches, which is equivalent to 79 ft 6 in. ANSWER: 79 ft 6 in. eSolutions Manual - Powered by Cognero Page 3 8-1 Geometric Mean Find the geometric mean between each pair of 11. 36 and 24 numbers. SOLUTION: 8. 81 and 4 By the definition, the geometric mean x of any two SOLUTION: numbers a and b is given by By the definition, the geometric mean x of any two Therefore, the geometric mean of 36 and 24 is numbers a and b is given by Therefore, the geometric mean of 81 and 4 is ANSWER: ANSWER: 18 12. 12 and 2.4 SOLUTION: 9. 25 and 16 By the definition, the geometric mean x of any two SOLUTION: numbers a and b is given by By the definition, the geometric mean x of any two Therefore, the geometric mean of 12 and 2.4 is numbers a and b is given by Therefore, the geometric mean of 25 and 16 is ANSWER: ANSWER: 20 13. 18 and 1.5 10. 20 and 25 SOLUTION: SOLUTION: By the definition, the geometric mean x of any two By the definition, the geometric mean x of any two numbers a and b is given by numbers a and b is given by Therefore, the geometric mean of 18 and 1.5 is Therefore, the geometric mean of 20 and 25 is ANSWER: ANSWER: eSolutions Manual - Powered by Cognero Page 4 8-1 Geometric Mean Write a similarity statement identifying the three similar triangles in the figure. 15. 14. SOLUTION: If the altitude is drawn to the hypotenuse of a right SOLUTION: triangle, then the two triangles formed are similar to If the altitude is drawn to the hypotenuse of a right the original triangle and to each other. triangle, then the two triangles formed are similar to the original triangle and to each other. is the altitude to the hypotenuse of the right is the altitude to the hypotenuse of the right triangle MNO. Therefore, triangle XYW. Therefore, ANSWER: ANSWER: eSolutions Manual - Powered by Cognero Page 5 8-1 Geometric Mean 16. 17. SOLUTION: SOLUTION: If the altitude is drawn to the hypotenuse of a right If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to triangle, then the two triangles formed are similar to the original triangle and to each other. the original triangle and to each other. is the altitude to the hypotenuse of the right triangle QRS. Therefore, ANSWER: is the altitude to the hypotenuse of the right triangle HGF. Therefore, ANSWER: Find x, y, and z. 18. SOLUTION: By the Geometric Mean (Altitude) Theorem the eSolutions Manual - Powered by Cognero Page 6 8-1 Geometric Mean altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of this altitude is the geometric mean between the lengths of these two segments. Solve for x. ANSWER: x = 6; ; By the Geometric Mean (Leg) Theorem the altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of a leg of this triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg. 19. Solve for y. SOLUTION: By the Geometric Mean (Altitude) Theorem the altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The By the Geometric Mean (Leg) Theorem the altitude length of this altitude is the geometric mean between drawn to the hypotenuse of a right triangle separates the lengths of these two segments. the hypotenuse into two segments. The length of a leg of this triangle is the geometric mean between the Solve for y. length of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Solve for z. eSolutions Manual - Powered by Cognero Page 7 8-1 Geometric Mean ANSWER: ; ; By the Geometric Mean (Leg) Theorem the altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of a leg of this triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg. So, 20. Solve for x. SOLUTION: By the Geometric Mean (Altitude) Theorem the altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments and the By the Geometric Mean (Leg) Theorem the altitude length of this altitude is the geometric mean between drawn to the hypotenuse of a right triangle separates the lengths of these two segments. the hypotenuse into two segments. The length of a leg of this triangle is the geometric mean between the Solve for y. length of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Solve for z. eSolutions Manual - Powered by Cognero Page 8 8-1 Geometric Mean By the Geometric Mean (Leg) Theorem the altitude drawn to the hypotenuse of a right triangle separates ANSWER: the hypotenuse into two segments and the length of a leg of this triangle is the geometric mean between the ; ; length of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Solve for x. 21. SOLUTION: By the Geometric Mean (Leg) Theorem the altitude drawn to the hypotenuse of a right triangle separates By the Geometric Mean (Leg) Theorem the altitude the hypotenuse into two segments and the length of a drawn to the hypotenuse of a right triangle separates leg of this triangle is the geometric mean between the the hypotenuse into two segments and the length of a length of the hypotenuse and the segment of the leg of this triangle is the geometric mean between the hypotenuse adjacent to that leg. length of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Solve for z. Solve for y. eSolutions Manual - Powered by Cognero Page 9 8-1 Geometric Mean By the Geometric Mean (Leg) Theorem the altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments and the length of a ANSWER: leg of this triangle is the geometric mean between the x ≈ 4.7; y ≈ 1.8; z ≈ 13.1 length of the hypotenuse and the segment of the hypotenuse adjacent to that leg.