Cross multiplying gives x2 = ab, so x = √ ab Find the geometric mean between each pair of numbers a
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Cross multiplying gives x2 = ab, so x = √ ab Find the geometric mean between each pair of numbers a
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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 42 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 25. 12 and 20 6. 4 and 25
7. 16 and 30 8. 10 and 100
9. 1 2 and 1 410. 17 and 3
11. 4 and 16 12. 3 and 24
8-1Example
424 or 2
6 ≈ 4.9
240 or 4
?? 15 ≈ 15.51000 or 10
10 ≈ 31.610
54 or 3
6≈ 7.31
480 or 4
30 ≈ 21.9
51 ≈ 7.1
872 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 25Study 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 CDIn ?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 = x225 = 25x Square each side.
9 x Divide each side by 25. Then y = RP - SP 259 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 81x
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 226. xzy 62
x = 2; x = 2; x =