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.