8-1 Similarity in Right Triangles

Objectives

Use geometric mean to find segment lengths in right triangles.

Apply similarity relationships in right triangles to solve problems.

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8-1 Similarity in Right Triangles

Vocabulary

geometric mean

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8-1 Similarity in Right Triangles

The geometric mean of two positive numbers is the positive square root of their product. Consider the proportion . In this case, the means of the proportion are the same number, and that number (x) is the geometric mean of the extremes.

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8-1 Similarity in Right Triangles Example 1A: Finding Geometric Means Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.

4 and 9

Let x be the geometric mean.

x

2 4

x



x

9  36

Def. of geometric mean Cross multiply

x = 6

Find the positive square root.

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8-1 Similarity in Right Triangles Example 1b: Finding Geometric Means Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.

6 and 15

Let x be the geometric mean.

6

x x

2 

x

15  90

Def. of geometric mean Cross multiply x

 3 10

Find the positive square root.

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8-1 Similarity in Right Triangles Example 1c: Finding Geometric Means Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.

2 and 8

Let x be the geometric mean.

x

2

x

2 

x

8  16

Def. of geometric mean Cross multiply x

 4

Find the positive square root.

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8-1 Similarity in Right Triangles

In a right triangle, an altitude drawn from the vertex of the right angle to the hypotenuse forms two right triangles.

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8-1 Similarity in Right Triangles Example: Identifying Similar Right Triangles Write a similarity statement comparing the three triangles.

Sketch the three right triangles with the angles of the triangles in corresponding positions.

W Z

By Theorem 8-1-1, ∆UVW ~ ∆UWZ ~ ∆WVZ.

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You can use Theorem 8-1-1 to write proportions comparing the side lengths of the triangles formed by the altitude to the hypotenuse of a right triangle.

All the relationships in red involve geometric means.

8-1 Similarity in Right Triangles

Example 1

h h

h

2

h

2   10

h

20

h

 20

h

 2 5

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8-1 Similarity in Right Triangles

Example 2

h h 12 12

b



b

10

b

2  120

b

 120

b

 2 30

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8-1 Similarity in Right Triangles

Example 2

h h 30 

b b

27

b

2  810

b

 810

b

 9 10

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8-1 Similarity in Right Triangles Helpful Hint

Once you’ve found the unknown side lengths, you can use the Pythagorean Theorem to check your answers.

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