8-1 similarity in right triangles - scott county schools and 1-30...holt mcdougal geometry ... holt...

Holt McDougal Geometry

8-1 Similarity in Right Triangles

Bellringer:1. Write a similarity statement

comparing the two triangles.

Simplify.

2. 3.

Solve each equation.

4. 5. 2x2 = 50 ±5

Holt McDougal 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 McDougal Geometry

8-1 Similarity in Right Triangles

geometric mean

Vocabulary

Holt McDougal 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 McDougal Geometry

8-1 Similarity in Right Triangles

Holt McDougal Geometry

8-1 Similarity in Right Triangles

Example 1: Identifying Similar Right Triangles

Write a similarity statement comparing the three triangles.

Holt McDougal Geometry

8-1 Similarity in Right Triangles

Example 1B:

Write a similarity statement comparing the three triangles.

Holt McDougal Geometry

8-1 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 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

8-1 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

Holt McDougal Geometry

8-1 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.

Holt McDougal Geometry

8-1 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

8-1 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

8-1 Similarity in Right Triangles

Bellringer: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

Holt McDougal 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 McDougal 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

Holt McDougal Geometry

8-1 Similarity in Right Triangles

Holt McDougal Geometry

8-1 Similarity in Right Triangles

Example 3: Finding Side Lengths in Right Triangles

Find x, y, and z.

Holt McDougal Geometry

8-1 Similarity in Right Triangles

Check It Out! Example 3

Find u, v, and w.