Find the Geometric Mean Between Each Pair of 4

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.

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 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.

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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:

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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:

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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 (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.

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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.

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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.

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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.

Use the value of y to solve the second proportion.

22.

SOLUTION:

the lengths of these two segments. By the Geometric Mean (Altitude) Theorem the

altitude drawn to the hypotenuse of a right triangle Solve for x. separates the hypotenuse into two segments and the length of this altitude is the geometric mean between the lengths of these two segments. So, Solve for x.

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Solve for y.

23.

SOLUTION:

By the Geometric Mean (Leg) Theorem the altitude By the Geometric Mean (Leg) Theorem the altitude drawn to the hypotenuse of a right triangle separates drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments and the length of a the hypotenuse into two segments and the length of a leg of this triangle is the geometric mean between the leg of this triangle is the geometric mean between the length of the hypotenuse and the segment of the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg. hypotenuse adjacent to that leg. Solve for z. Solve for z.

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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 leg of this triangle is the geometric mean between the ANSWER: length of the hypotenuse and the segment of the ; ;z = 32 hypotenuse adjacent to that leg. 24. MODELING Evelina is hanging silver stars from Use the value of z to solve the second proportion for the gym ceiling using string for the homecoming x. dance. She wants the ends of the strings where the stars will be attached to be 7 feet from the floor. Use the diagram to determine how long she should make the strings.

By the Geometric Mean (Altitude) Theorem the SOLUTION: altitude drawn to the hypotenuse of a right triangle Let x represent the length of the string. Since the star separates the hypotenuse into two segments and the will be 7 feet from the floor, x + 7 is the total length length of this altitude is the geometric mean between of string to floor. Since we are given 5 feet from the the lengths of these two segments. floor in the diagram. The distance to the 5 ft point will be x + 2. Solve for y.

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25. MODELING Makayla is using a book to sight the top of a waterfall. Her eye level is 5 feet from the ground and she is a horizontal distance of 28 feet from the waterfall. Find the height of the waterfall to the nearest tenth of a foot.

Use the Geometric Mean (Altitude) Theorem to find x.

SOLUTION: We have the diagram as shown.

So she should make the strings of length 18 feet.

ANSWER: By the Geometric Mean (Altitude) Theorem the altitude drawn to the hypotenuse of a right triangle 18 ft separates the hypotenuse into two segments. The length of this altitude is the geometric mean between the lengths of these two segments.

So, the total height of the waterfall is 156.8 + 5 = 161.8 ft.

ANSWER: 161.8 ft

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Find the geometric mean between each pair of 28. and numbers.

26. and 60 SOLUTION: 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 numbers a and b is given by Therefore, the geometric mean of is Therefore, the geometric mean of and 60 is

ANSWER:

ANSWER: or 2.2 or 3.5 Find x, y, and z.

27. and

SOLUTION: By the definition, the geometric mean x of any two numbers a and b is given by

Therefore, the geometric mean of is 29. SOLUTION:

ANSWER:

or 0.8

Solve for x.

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Use the value of x to solve the for z.

ANSWER:

; ; z = 3

30.

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 length of this altitude is the geometric mean between the lengths of these two segments. By the Geometric Mean (Altitude) Theorem the Solve for y. altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments and the length of this altitude is the geometric mean between the lengths of these two segments.

Solve for z. eSolutions Manual - Powered by Cognero Page 15 8-1 Geometric Mean

Solve for y.

Solve for x.

ANSWER:

; ;

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31. ALGEBRA The geometric mean of a number and four times the number is 22. What is the number?

SOLUTION: Let x be the first number. Then the other number will be 4x. By the definition, the geometric mean x of any two numbers a and b is given by So,

Therefore, the number is 11.

ANSWER: 11 By the Geometric Mean (Leg) Theorem the altitude Use similar triangles to find the value of x. drawn to the hypotenuse of a right triangle separates 32. Refer to the figure on page 543. 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. Let y be the shorter segment of the hypotenuse of the bigger right triangle.

So, the shorter segment of the right triangle is about SOLUTION: 14.2-4.1=10.1 ft.

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So, the longer segment of the right triangle is about 3.98 ft.

33. Refer to the figure on page 543.

The length of the segment is about 3.5 ft.

ANSWER: 3.5 ft

34. Refer to the figure on page 543.

SOLUTION:

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 leg of this triangle is the geometric mean between the By the Geometric Mean (Leg) Theorem the altitude length of the hypotenuse and the segment of the drawn to the hypotenuse of a right triangle separates hypotenuse adjacent to that leg. Let y be the shorter the hypotenuse into two segments and the length of a segment of the hypotenuse of the bigger right leg of this triangle is the geometric mean between the triangle. length of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Let y be the shorter segment of the hypotenuse of the bigger right triangle.

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ALGEBRA Find the value of the variable.

35.

SOLUTION: By the Geometric Mean (Altitude) Theorem the altitude drawn to the hypotenuse of a right triangle So, the longer segment of the right triangle is about separates the hypotenuse into two segments and the 25.9 ft. length of this altitude is the geometric mean between the lengths of these two segments. By the Geometric Mean (Altitude) Theorem the altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments and the length of this altitude is the geometric mean between the lengths of these two segments.

Then x is about 13.5 ft.

ANSWER: 13.75 ft

ANSWER: 5

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36. 37. SOLUTION: By the Geometric Mean (Altitude) Theorem the SOLUTION: altitude drawn to the hypotenuse of a right triangle By the Geometric Mean (Altitude) Theorem the separates the hypotenuse into two segments and the altitude drawn to the hypotenuse of a right triangle length of this altitude is the geometric mean between separates the hypotenuse into two segments and the the lengths of these two segments. length of this altitude is the geometric mean between the lengths of these two segments.

Use the quadratic formula to find the roots of the quadratic equation.

If w = –16, the length of the altitude will be –16 + 4 = –12 which is not possible, as a length cannot be negative. Therefore, w = 8. Since m is a length, it cannot be negative. Therefore, ANSWER: m = 4. 8 ANSWER: 4

38. CONSTRUCTION A room-in-attic truss is a truss design that provides support while leaving area that eSolutions Manual - Powered by Cognero Page 20 8-1 Geometric Mean

can be enclosed as living space. In the diagram, So, the total length AE is about (96.3 + 60 + 84.5) in. BCA and EGB are right angles, is = 240.8 in. or 20.07 ft. isosceles, is an altitude of , and is an ANSWER: altitude of . If DB = 5 feet, CD = 6 feet 4 about 20.07 ft inches, BF = 10 feet 10 inches, and EG = 4 feet 6 inches, what is AE? CONSTRUCT ARGUMENTS Write a proof for each theorem. 39. Theorem 8.1

SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given a right angle and an altitude of a triangle. Use the properties that you have SOLUTION: learned about congruent segments, altitudes, right triangles, and equivalent expressions in algebra to walk through the proof.

Given: PQR is a right angle. is an altitude of . Prove:

First find Given that is isosceles, then bisects Since is 10 ft 10 in.or 130 in., then BG = GF = Proof: 65 in. Statements (Reasons) is a right triangle and by the Pythagorean 1. PQR is a right angle. is an altitude of Theorem, . (Given) 2. (Definition of altitude) 3. 1 and 2 are right angles. (Definition of perpendicular lines) 4. ; (All right angles are Next find AD. Let AD = x. congruent.) By the Geometric Mean (Altitude) Theorem the 5. ; (Congruence of angles is altitude drawn to the hypotenuse of a right triangle reflexive.) separates the hypotenuse into two segments and the 6. ; AA (Similarity length of this altitude is the geometric mean between the lengths of these two segments. Statements 4 and 5) 7. (Similarity of triangles is transitive.)

ANSWER: Given: PQR is a right angle. is an altitude of eSolutions Manual - Powered by Cognero Page 21 8-1 Geometric Mean

. from the vertex of the right angle to the hypotenuse Prove: of a right triangle, then the two triangles formed are similar to the given triangle and to each other. So

by definition of similar triangles. Proof: ANSWER: Statements (Reasons) 1. PQR is a right angle. is an altitude of Given: is a right triangle. is an altitude of . (Given) . 2. (Definition of altitude) 3. 1 and 2 are right s. (Definition of lines) 4. ; (All right s are .) 5. ; (Congruence of angles is reflexive.) 6. ; AA (Similarity Prove: Statements 4 and 5) 7. (Similarity of triangles is Proof: It is given that is a right triangle and transitive.) is an altitude of . ADC is a right angle by the definition of a right triangle. Therefore, 40. Theorem 8.2 , because if the altitude is drawn from the vertex of the right angle to the hypotenuse SOLUTION: of a right triangle, then the two triangles formed are You need to walk through the proof step by step. similar to the given triangle and to each other. So Look over what you are given and what you need to by definition of similar triangles. prove. Here, you are given a right triangle and an altitude. Use the properties that you have learned about right triangles, altitudes, congruent segment,s 41. Theorem 8.3 and equivalent expressions in algebra to walk through SOLUTION: the proof. You need to walk through the proof step by step. Look over what you are given and what you need to Given: is a right triangle. is an altitude of prove. Here, you are given a right triangle and an . altitude. Use the properties that you have learned about congruent segments, right triangles, altitudes, and equivalent expressions in algebra to walk through the proof. Given: ADC is a right angle. is an altitude of .

Prove: Proof: It is given that is a right triangle and is an altitude of . ADC is a right angle by the definition of a right triangle. Therefore, , because if the altitude is drawn eSolutions Manual - Powered by Cognero Page 22 8-1 Geometric Mean

top of the subject, the camera, and the bottom of the Prove: ; subject is called the viewing angle, as shown in the Proof: diagram. Natalie is taking a picture of Bigfoot #5, Statements (Reasons) which is 15 feet 6 inches tall. She sets her camera on a tripod that is 5 feet above ground level. The vertical 1. ADC is a right angle. is an altitude of viewing angle of her camera is set for . (Given) 2. is a right triangle. (Definition of right triangle) 3. ; (If the altitude is drawn from the vertex of the right angle to the hypotenuse of a right triangle, then the 2 triangles formed are similar to the given triangle and to each other.) a. Sketch a diagram of this situation. b. How far away from the truck should Natalie stand 4. ; (Definition of similar so that she perfectly frames the entire height of the triangles) truck in her shot?

ANSWER: SOLUTION: Given: ADC is a right angle. is an altitude of a. We are given the height of the truck and the height . of the camera.

Prove: ; b. Proof: Statements (Reasons) 1. ADC is a right angle. is an altitude of (Given) 2. is a right triangle. (Definition of right triangle) 3. ; (If the altitude is drawn from the vertex of the rt. to the hypotenuse of a rt. , then the 2 s formed are similar to the given and to each other.)

4. ; (Definition of similar triangles) By the Geometric Mean (Altitude) Theorem the 42. TRUCKS In photography, the angle formed by the altitude drawn to the hypotenuse of a right triangle eSolutions Manual - Powered by Cognero Page 23 8-1 Geometric Mean

separates the hypotenuse into two segments and the 44. PROOF Derive the Pythagorean Theorem using the length of this altitude is the geometric mean between figure at the right and the Geometric Mean (Leg) the lengths of these two segments. Theorem.

Therefore, she should stand about 7.2 ft away from the truck. SOLUTION: ANSWER: You need to walk through the proof step by step. a. Look over what you are given and what you need to prove. Use the properties that you have learned about congruent, right triangles, altitudes, and equivalent expressions in algebra to walk through the proof. Using the Geometric Mean (Leg) Theorem, and . Squaring both values, a2 = 2 2 2 b. about 7.2 ft yc and b = xc. The sum of the squares is a + b = yc + xc. Factoring the c on the right side of the 43. FINANCE The average rate of return on an 2 2 investment over two years is the geometric mean of equation, a + b = c(y + x). By the Segment the two annual returns. If an investment returns 12% Addition Postulate, c = y + x. Substituting, a2 + b2 = one year and 7% the next year, what is the average c(c) or a2 + b2 = c2. rate of return on this investment over the two-year period? ANSWER: SOLUTION: Using the Geometric Mean (Leg) Theorem, By the definition, the geometric mean x of any two and . Squaring both values, a2 = numbers a and b is given by yc and b2 = xc. The sum of the squares is a2 + b2 = Therefore, the geometric mean of 7% and 12% is yc + xc. Factoring the c on the right side of the equation, a2 + b2 = c(y + x). By the Segment Addition Postulate, c = y + x. Substituting, a2 + b2 = ANSWER: c(c) or a2 + b2 = c2. about 9%

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Determine whether each statement is always, 46. The geometric mean for two perfect squares is a sometimes, or never true. Explain your positive integer. reasoning. SOLUTION: 45. The geometric mean for consecutive positive integers is the mean of the two numbers. The square root of a perfect square is always a positive integer. Therefore if you multiply two perfect SOLUTION: squares, the square root will always be a positive Let x be the first number. Then x + 1 is the next. The integer. Then since is equal to , the geometric mean of two consecutive integers is geometric mean for two perfect squares will always . be the product of two positive integers, which is a positive integer. Thus, the statement is always true. The mean of the two number is . Consider the example where a is 16 and b is 25. Set the two numbers equal. Then . ANSWER: Always; sample answer: Since is equal to , the geometric mean for two perfect squares will always be the product of two positive integers, which is a positive integer.

47. The geometric mean for two positive integers is = another integer. If you set the two expressions equal to each other, SOLUTION: the equation has no real solution. Therefore, the When the product of the two integers is a perfect statement is never true. square, the geometric mean will be a positive integer. Therefore, the statement is sometimes true. ANSWER:

Never; sample answer: The geometric mean of two For example, 6 is the geometric mean between 4 and consecutive integers is , and the average of 9 while is the geometric mean between 5 and 8. two consecutive integers is . If you set the two expressions equal to each other, the equation has no solution.

ANSWER: Sometimes; sample answer: When the product of the two integers is a perfect square, the geometric mean will be a positive integer.

48. MULTIPLE REPRESENTATIONS In this problem, you will investigate geometric mean. a. Tabular Copy and complete the table of five ordered pairs (x, y) such that . eSolutions Manual - Powered by Cognero Page 25 8-1 Geometric Mean

increases. So, the graph that would be formed will be a hyperbola.

ANSWER: a.

b. Graphical Graph the ordered pairs from your table in a scatter plot. c. Verbal Make a conjecture as to the type of graph that would be formed if you connected the points from your scatter plot. Do you think the graph of any set of ordered pairs that results in the same geometric b. mean would have the same general shape? Explain your reasoning.

SOLUTION: a. Find pairs of numbers with product of 64. It is easier to graph these points, if you choose integers.

c. hyperbola; yes; As x increases, y decreases, and as x decreases, y increases.

c. What relationship do you notice happening between the x and the y values? Pay close attention to what happens to the y-values, as the x's increase. As x increases, y decreases, and as x decreases, y eSolutions Manual - Powered by Cognero Page 26 8-1 Geometric Mean

49. ERROR ANALYSIS Aiden and Tia are finding the leg of this triangle is the geometric mean between the value x in the triangle shown. Is either of them length of the hypotenuse and the segment of the correct? Explain your reasoning. hypotenuse adjacent to that leg.

Solve the proportion for x.

Use the quadratic formula to find the roots of the quadratic equation.

SOLUTION: 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. Since x is a length, it cannot be negative. Therefore, x is about 5.2.

By the Geometric Mean (Leg) Theorem the altitude drawn to the hypotenuse of a right triangle separates Therefore, neither of them are correct. the hypotenuse into two segments and the length of a leg of this triangle is the geometric mean between the ANSWER: length of the hypotenuse and the segment of the Neither; sample answer: On the similar triangles hypotenuse adjacent to that leg. created by the altitude, the leg that is x units long on the smaller triangle corresponds with the leg that is 8 units long on the larger triangle, so the correct Use the value of x to solve the proportion. proportion is and x is about 5.7.

50. CHALLENGE Refer to the figure. Find x, y, and z.

By the Geometric Mean (Altitude) Theorem the SOLUTION: altitude drawn to the hypotenuse of a right triangle By the Geometric Mean (Leg) Theorem the altitude separates the hypotenuse into two segments and the drawn to the hypotenuse of a right triangle separates length of this altitude is the geometric mean between the hypotenuse into two segments and the length of a the lengths of these two segments. eSolutions Manual - Powered by Cognero Page 27 8-1 Geometric Mean

52. REASONING Refer to the figure. The orthocenter of is located 6.4 units from point D. Find BC.

ANSWER: x = 5.2, y = 6.8, z = 11

51. OPEN-ENDED Find two pairs of whole numbers with a geometric mean that is also a whole number. SOLUTION: What condition must be met in order for a pair of From the figure, B is the orthocenter of the triangle numbers to produce a whole-number geometric ABC since ABC is a right triangle. mean? Therefore BD = 6.4. SOLUTION: Find two pairs of whole numbers with a geometric Use the Pythagorean Theorem to find AB. mean that is also a whole number are : 9 and 4, 8 and 8. In order for two whole numbers to result in a whole- number geometric mean, their product must be a perfect square. and . 36 and 64 are both perfect squares. Let CD = x. Use the Geometric Mean (Altitude) Theorem to find x. ANSWER: Sample answer: 9 and 4, 8 and 8; In order for two whole numbers to result in a whole-number geometric mean, their product must be a perfect square. Use the Pythagorean Theorem to find BC.

ANSWER: 10.0

53. WRITING IN MATH Compare and contrast the arithmetic and geometric means of two numbers. When will the two means be equal? Justify your reasoning.

SOLUTION: When comparing these two means, consider the commonalities and differences of their formulas, when finding the means of two numbers such as a eSolutions Manual - Powered by Cognero Page 28 8-1 Geometric Mean

and b. 54. The figure shows the paths in a botanical garden.

Both the arithmetic and the geometric mean calculate a value between two given numbers. The arithmetic

mean of two numbers a and b is , and the

geometric mean of two numbers a and b is . The two means will be equal when a = b. What is the distance from the greenhouse to the

pond? Justification: A 10.24 m B 35.24 m C 39.06 m D 64.06 m

SOLUTION: 16 is the geometric mean between 25 and RP (the distance from the rose beds to the pond).

ANSWER: Sample answer: Both the arithmetic and the So the distance from the greenhouse to the pond is 25 geometric mean calculate a value between two given + 10.24 = 35.24, so choice B is correct. numbers. The arithmetic mean of two numbers a and ANSWER: b is , and the geometric mean of two numbers B a and b is . The two means will be equal when a = b. Justification:

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55. What is the area of ΔJKL? 56. Which of the following is the best estimate of the length of ?

A 10 ft² A 37.9 B 50 ft² B 15.7 C 125 ft² C 14.5 D 250 ft² D 8.9

SOLUTION: SOLUTION: To determine the area of ΔJKL, you will need a base Use the Geometric Mean (Leg) Theorem to set up a and a height to that base of the triangle. We have the proportion to find the length of . length of and can determine the height to that base ( ) by using the Geometric Mean (Altitude) Theorem.

Find height :

. Therefore, the correct choice is B.

ANSWER: B

10 = KM

The area of ΔJKL = . The correct choice is C.

ANSWER: C

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57. Zachary wants to estimate the height of a cliff. He 58. In the figure, is perpendicular to , and is stands 15 feet from the base of the cliff. perpendicular to . What is BC?

A If his eye level is at 5.5 feet, which of the following is B closest to the height of the cliff? C A 10.6 ft D B 17.5 ft C 40.9 ft SOLUTION: D 46.4 ft Use the Geometric Mean (Leg) Theorem to set up a proportion to solve for BC, using the diagram below. SOLUTION: Use the Geometric Mean (Altitude) Theorem to determine the value of x in the drawing below.

The correct choice is D.

ANSWER: D

The height of the cliff would be 40.9 + 5.5 = 46.4 ft. The correct choice is D.

ANSWER: D

eSolutions Manual - Powered by Cognero Page 31 8-1 Geometric Mean

59. MULTI-STEP An altitude is drawn to the hypotenuse of a right triangle, separating the hypotenuse into two segments. The segments of the hypotenuse are in the ratio 4 : 9.

a. What is the ratio of the length of the shorter segment of the hypotenuse to the length of the altitude?

b. Suppose the length of the altitude is 42 inches. What are the lengths of the hypotenuse segments?

SOLUTION: a. The geometric mean of 4 and 9 is 6, so the altitude is 6. So the ratio of the shorter segment of the hypotenuse to the altitude is .

b. Let x represent the length of the shorter segment of the hypotenuse. Then

So the shorter segment is 28 inches.

Let y represent the length of the longer segment of the hypotenuse. Then

So the longer segment is 63 inches.

ANSWER: 28 inches; 63 inches