Holt Geometry

8-1 Similarity in Right Triangles

Use geometric mean to find segment lengths in right triangles.

Apply similarity relationships in right triangles to solve problems.

Objectives

Holt Geometry

8-1 Similarity in Right Triangles

geometric mean

Vocabulary

Holt Geometry

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.

Holt Geometry

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.

Def. of geometric mean

x = 6 Find the positive square root.

4

9

x

x

Cross multiply2 36x

Holt Geometry

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.

Def. of geometric mean

Find the positive square root.

6

15

x

x

Cross multiply2 90x

3 10x

Holt Geometry

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.

Def. of geometric mean

Find the positive square root.

2

8

x

x

Cross multiply2 16x

4x

Holt Geometry

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.

Holt Geometry

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.

By Theorem 8-1-1, ∆UVW ~ ∆UWZ ~ ∆WVZ.

Z

W

Holt Geometry

8-1 Similarity in Right Triangles

Holt Geometry

8-1 Similarity in Right Triangles

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.

Holt Geometry

8-1 Similarity in Right Triangles

Example 1

10

2

h

h

2 20h c

a b

nm

h

m = 2, n = 10, h = ___

c

a b

nm

h

c

a

b

n

m

hb

n

ha

m

h

2 10

20

2 5

h

h

2 10

Holt Geometry

8-1 Similarity in Right Triangles

Example 2

12

10

b

b

2 120b

c

a b

nm

h

m = 2, n = 10, b = ___

c

a b

nm

h

c

a

b

n

m

hb

n

ha

m

h

2 10

120

2 30

b

b

12

2 10

Holt Geometry

8-1 Similarity in Right Triangles

Example 2

30

27

b

b

2 810b

c

a b

nm

h

n = 27, c = 30, b = ___

c

a b

nm

h

c

a

b

n

m

hb

n

ha

m

h

27 30

810

9 10

b

b

27

Holt Geometry

8-1 Similarity in Right Triangles

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

Helpful Hint