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Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.

Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.

Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.

9.1 Similarity in Right Triangles

Warm Up

1. Write a similarity statement

comparing the two triangles.

Simplify.

2. 3.

Solve each equation.

4. 5. 2x2 = 50

∆ADB ~ ∆EDC

±5

9.1 Similarity in Right Triangles

Objectives

Use geometric mean to find segment lengths in right triangles.

Apply similarity relationships in right triangles to solve problems.

9.1 Similarity in Right Triangles

9.1 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.

9.1 Similarity in Right Triangles

W

Z

Example 1: 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.

9.1 Similarity in Right Triangles

Check It Out! Example 1

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, ∆LJK ~ ∆JMK ~ ∆LMJ.

9.1 Similarity in Right Triangles

Consider the proportion . In this case, the means of the proportion are the same number, and that number is the geometric mean of the extremes.

The geometric meanof two positive numbers is the positive square root of their product. So the geometric mean of a and b is the positive number x such

that , or x2 = ab.

9.1 Similarity in Right Triangles

9.1 Similarity in Right Triangles

Example 2A: Finding Geometric Means

Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.

4 and 25

Let x be the geometric mean.

x2 = (4)(25) = 100

Def. of geometric mean

x = 10

Find the positive square root.

9.1 Similarity in Right Triangles

Example 2B: Finding Geometric Means

Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.

5 and 30

Let x be the geometric mean.

x2 = (5)(30) = 150

Def. of geometric mean

Find the positive square root.

9.1 Similarity in Right Triangles

Check It Out! Example 2a

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.

x2 = (2)(8) = 16

Def. of geometric mean

x = 4

Find the positive square root.

9.1 Similarity in Right Triangles

Check It Out! Example 2b

10 and 30

Let x be the geometric mean.

x2 = (10)(30) = 300

Def. of geometric mean

Find the positive square root.

9.1 Similarity in Right Triangles

Check It Out! Example 2c

8 and 9

Let x be the geometric mean.

x2 = (8)(9) = 72

Def. of geometric mean

Find the positive square root.

9.1 Similarity in Right Triangles

You can use Geometric Means 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.

9.1 Similarity in Right Triangles

Example 3: Finding Side Lengths in Right Triangles

Find x, y, and z.

62 = (9)(x)

6 is the geometric mean of 9 and x.

x = 4

Divide both sides by 9.

y is the geometric mean of 4 and 13.

y2 = (4)(13) = 52

Find the positive square root.

z2 = (9)(13) = 117

z is the geometric mean of 9 and 13.

Find the positive square root.

9.1 Similarity in Right Triangles

Helpful Hint

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

9.1 Similarity in Right Triangles

Check It Out! Example 3

Find u, v, and w.

92 = (3)(u) 9 is the geometric mean of

u and 3.

u = 27 Divide both sides by 3.

w2 = (27 + 3)(27) w is the geometric mean of

u + 3 and 27.

Find the positive square root.

v2 = (27 + 3)(3) v is the geometric mean of

u + 3 and 3.

Find the positive square root.

9.1 Similarity in Right Triangles

Example 4: Measurement Application

To estimate the height of a Douglas fir, Jan positions herself so that her lines of sight to the top and bottom of the tree form a 90º angle. Her eyes are about 1.6 m above the ground, and she is standing 7.8 m from the tree. What is the height of the tree to the nearest meter?

9.1 Similarity in Right Triangles

Example 4 Continued

Let x be the height of the tree above eye level.

7.8 is the geometric mean of 1.6 and x.

(7.8)2 = 1.6x

x = 38.025 ≈ 38

Solve for x and round.

The tree is about 38 + 1.6 = 39.6, or 40 m tall.

9.1 Similarity in Right Triangles

Check It Out! Example 4

A surveyor positions himself so that his line of sight to the top of a cliff and his line of sight to the bottom form a right angle as shown.

What is the height of the cliff to the nearest foot?

9.1 Similarity in Right Triangles

Check It Out! Example 4 Continued

Let x be the height of cliff above eye level.

(28)2 = 5.5x

28 is the geometric mean of 5.5 and x.

x 142.5

Divide both sides by 5.5.

The cliff is about 142.5 + 5.5, or 148 ft high.

9.1 Similarity in Right Triangles

Lesson Quiz: Part I

1. 8 and 18

2. 6 and 15

12

9.1 Similarity in Right Triangles

Lesson Quiz: Part II

For Items 3–6, use ∆RST.

3. Write a similarity statement comparing the three triangles.

4. If PS = 6 and PT = 9, find PR.

5. If TP = 24 and PR = 6, find RS.

6. Complete the equation (ST)2 = (TP + PR)(?).

∆RST ~ ∆RPS ~ ∆SPT

4

TP

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