# GEOMETRY 10-3 Similarity in Right Triangles Warm Up Warm Up Lesson Presentation Lesson Presentation...

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Slide 1comparing the two triangles.

Use geometric mean to find segment lengths in right triangles.

Apply similarity relationships in right triangles to solve problems.

Objectives

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.

10-3

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.

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.

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.

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

x2 = (4)(25) = 100

10-3

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

10-3

TEACH! Example 2a

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

2 and 8

x2 = (2)(8) = 16

10-3

TEACH! 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

10-3

TEACH! 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

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.

10-3

Find x, y, and z.

62 = (9)(x)

x = 4

y2 = (4)(13) = 52

Find the positive square root.

z2 = (9)(13) = 117

Find the positive square root.

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

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

u + 3 and 27.

u and 3.

Find the positive square root.

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

u + 3 and 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?

10-3

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.

10-3

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?

10-3

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

Divide both sides by 5.5.

x 142.5

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

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

Use geometric mean to find segment lengths in right triangles.

Apply similarity relationships in right triangles to solve problems.

Objectives

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.

10-3

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.

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.

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.

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

x2 = (4)(25) = 100

10-3

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

10-3

TEACH! Example 2a

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

2 and 8

x2 = (2)(8) = 16

10-3

TEACH! 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

10-3

TEACH! 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

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.

10-3

Find x, y, and z.

62 = (9)(x)

x = 4

y2 = (4)(13) = 52

Find the positive square root.

z2 = (9)(13) = 117

Find the positive square root.

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

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

u + 3 and 27.

u and 3.

Find the positive square root.

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

u + 3 and 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?

10-3

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.

10-3

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?

10-3

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

Divide both sides by 5.5.

x 142.5

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

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

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