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Similarity in Right Triangles

Holt Geometry

Warm Up

Lesson Presentation

Lesson Quiz

Holt McDougal Geometry

Holt McDougal Geometry

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Similarity in Right Triangles

Warm Up

1. Write a similarity statement

comparing the two triangles.

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Simplify.

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2. 3.

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Solve each equation.

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4. 5. 2x² = 50

∆ADB ~ ∆EDC

±5

Holt McDougal Geometry

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Similarity in Right Triangles

Use geometric mean to find segment lengths in right triangles.

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Apply similarity relationships in right triangles to solve problems.

Objectives

Holt McDougal Geometry

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Similarity in Right Triangles

geometric mean

Vocabulary

Holt McDougal Geometry

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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 McDougal Geometry

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Similarity in Right Triangles

Holt McDougal Geometry

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Similarity in Right Triangles

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.

Holt McDougal Geometry

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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.

Holt McDougal Geometry

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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 mean of 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.

Holt McDougal Geometry

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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.

Holt McDougal Geometry

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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.

Let x be the geometric mean.

5 and 30

x² = (5)(30) = 150

Def. of geometric mean

Find the positive square root.

Holt McDougal Geometry

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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.

Holt McDougal Geometry

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Similarity in Right Triangles

Check It Out! Example 2b

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

Let x be the geometric mean.

10 and 30

x² = (10)(30) = 300

Def. of geometric mean

Find the positive square root.

Holt McDougal Geometry

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Similarity in Right Triangles

Check It Out! Example 2c

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

Let x be the geometric mean.

8 and 9

x² = (8)(9) = 72

Def. of geometric mean

Find the positive square root.

Holt McDougal Geometry

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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 McDougal Geometry

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Similarity in Right Triangles

Holt McDougal Geometry

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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.

y² = (4)(13) = 52

y is the geometric mean of 4 and 13.

Find the positive square root.

z² = (9)(13) = 117

z is the geometric mean of 9 and 13.

Find the positive square root.

Holt McDougal Geometry

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Similarity in Right Triangles

Holt McDougal Geometry

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Similarity in Right Triangles

Check It Out! Example 3

Find u, v, and w.

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

u + 3 and 27.

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

u and 3.

u = 27 Divide both sides by 3.

Find the positive square root.

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

u + 3 and 3.

Find the positive square root.

Holt McDougal Geometry

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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?

Holt McDougal Geometry

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Similarity in Right Triangles

Example 4 Continued

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

x = 38.025 ≈ 38

(7.8)² = 1.6x

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

7.8 is the geometric mean of 1.6 and x.

Solve for x and round.

Holt McDougal Geometry

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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?

Holt McDougal Geometry

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Similarity in Right Triangles

Check It Out! Example 4 Continued

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

Let x be the height of cliff above eye level.

(28)² = 5.5x

28 is the geometric mean of 5.5 and x.

Divide both sides by 5.5.

x ≈ 142.5

Holt McDougal Geometry

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Similarity in Right Triangles