Holt 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 Geometry

8-1 Similarity in Right Triangles

Holt Geometry

8-1 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.

Z

W

Holt 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 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

Let x be the geometric mean.

x2 = (4)(25) = 100 Def. of geometric mean

x = 10 Find the positive square root.

Holt 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 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 Geometry

8-1 Similarity in Right Triangles

Holt Geometry

8-1 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.

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 Geometry

8-1 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.