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Lesson 8-1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Compan ies, Inc.

NAME DATE PERIOD

Chapter 8 5 Glencoe Geometry

Study Guide and Intervention

Geometric Mean

Geometric Mean The geometric mean between two numbers is the positive square root of their product. For two positive numbers a and b , the geometric mean of a and b is the positive number x in the proportion a x x b . Cross multiplying gives x 2 = ab, so x = ab . Find the geometric mean between each pair of numbers. a. 12 and 3 x = ⎷ ab Definition of geometric mean = ⎷ ??? 12 . 3 a = 12 and b = 3 (2 . 2 . 3) . 3 Factor. = 6 Simplify. The geometric mean between 12 and 3 is 6.b. 8 and 4 x = ab Definition of geometric mean = ⎷ ?? 8 . 4 a = 8 and b = 4 (2 . 4) . 4 Factor. = ⎷ ??? 16 . 2 Associative Property = 4 ⎷ ? 2 Simplify.

The geometric mean between 8 and 4

is 4

2 or about 5.7.

Exercises

Find the geometric mean between each pair of numbers.

1. 4 and 4 2. 4 and 6

3. 6 and 9 4.

1 2 and 2

5. 12 and 20 6. 4 and 25

7. 16 and 30 8. 10 and 100

9. 1 2 and 1 4

10. 17 and 3

11. 4 and 16 12. 3 and 24

8-1

Example

4

24 or 2

6 ≈ 4.9

240 or 4

?? 15 ≈ 15.5

1000 or 10

10 ≈ 31.610

54 or 3

6≈ 7.31

480 or 4

30 ≈ 21.9

51 ≈ 7.1

8

72 or 6

2 ≈ 8.5

1 8 or 2 4 ≈ 0.4 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Compan ies, Inc.

NAME DATE PERIOD

Chapter 8 6 Glencoe Geometry

zy x 15 R QP S 25

Study Guide and Intervention (continued)

Geometric Mean

Geometric Means in Right Triangles In the diagram, ABC ≂ ? ADB ≂ ? BDC. An altitude to the hypotenuse of a right triangle forms two right triangles. The two triangles are similar and each is similar to the original triangle.

Use right ? ABC with

BD ?

AC . Describe two geometric

means. a. ? ADB ≂ ? BDC so AD BD BD CD

In ?ABC, the altitude is the geometric

mean between the two segments of the hypotenuse. b. ? ABC ≂ ? ADB and ? ABC ≂ ? BDC, so AC AB AB AD and AC BC BC DC In ?

ABC, each leg is the geometric

mean between the hypotenuse and the segment of the hypotenuse adjacent to that leg. Find x, y, and z. 15 =

RP ? SP Geometric Mean (Leg) Theorem

15 ⎷ ?? 25x RP = 25 and SP = x

225 = 25x Square each side.

9 x Divide each side by 25. Then y = RP - SP 25
9 16 z

RS ? RP Geometric Mean (Leg) Theorem

⎷ ??? 16 ? 25 RS = 16 and RP = 25 ⎷ ?? 400 Multiply.

20 Simplify.

Exercises

Find x, y, and z to the nearest tenth.

1. x 13 2. zx y52 3. zxy 81
x

3 ≈ 1.7 x =

10 ≈ 3.2; x = 3;

y =

14 ≈ 3.7; y =

72 or 6

2 ≈ 8.5;

z =

35 ≈ 5.9 z =

8 or 2

2 ≈ 2.8

4. xy 1 ⎷‾3 ⎷?12 5. xz y 22
6. xzy 62
x = 2; x = 2; x =

12 or 2

3 ≈ 3.5;

y = 3 y =

8 or 2

2 ≈ 2.8; y =

8 or 2

2 ≈ 2.8;

z =

8 or 2

2 ≈ 2.8 z =

24 or 2

6 ≈ 4.9

8-1 CDBA

Example 1Example 2

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