GEOMETRY

10-3 Similarity in Right Triangles

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

GEOMETRY

10-3 Similarity in Right Triangles

Warm Up1. Write a similarity statement

comparing the two triangles.

Simplify.

2. 3.

Solve each equation.

4. 5. 2x2 = 50

∆ADB ~ ∆EDC

±5

GEOMETRY

10-3 Similarity in Right Triangles

Use geometric mean to find segment lengths in right triangles.

Apply similarity relationships in right triangles to solve problems.

Objectives

GEOMETRY

10-3 Similarity in Right Triangles

geometric mean

Vocabulary

GEOMETRY

10-3 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.

GEOMETRY

10-3 Similarity in Right Triangles

GEOMETRY

10-3 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.

∆UVW ~ ∆UWZ ~ ∆WVZ.

Z

W

GEOMETRY

10-3 Similarity in Right Triangles

TEACH! Example 1

Write a similarity statement comparing the three triangles.

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

∆LJK ~ ∆JMK ~ ∆LMJ.

GEOMETRY

10-3 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.

GEOMETRY

10-3 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.

GEOMETRY

10-3 Similarity in Right Triangles

TEACH! 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.

GEOMETRY

10-3 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.

GEOMETRY

10-3 Similarity in Right Triangles

GEOMETRY

10-3 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.

GEOMETRY

10-3 Similarity in Right Triangles

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

Helpful Hint

GEOMETRY

10-3 Similarity in Right Triangles

TEACH! 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.

GEOMETRY

10-3 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?

GEOMETRY

10-3 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.

GEOMETRY

10-3 Similarity in Right Triangles

TEACH! 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?

GEOMETRY

10-3 Similarity in Right Triangles

TEACH! 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

GEOMETRY

10-3 Similarity in Right Triangles

Lesson Quiz: Part I

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

1. 8 and 18

2. 6 and 15

12

GEOMETRY

10-3 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