Use geometric mean to find segment lengths in right triangles.
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Transcript of Use geometric mean to find segment lengths in right triangles.
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
geometric mean
Vocabulary
Holt Geometry
8-1 Similarity in Right Triangles
The geometric mean of two positive numbers is the
positive square root of their product.
Consider the proportion . In this case, the
means of the proportion are the same number, and
that number (x) is the geometric mean of the extremes.
Holt Geometry
8-1 Similarity in Right Triangles
Example 1A: Finding Geometric Means
Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.
4 and 9
Let x be the geometric mean.
Def. of geometric mean
x = 6 Find the positive square root.
4
9
x
x
Cross multiply2 36x
Holt Geometry
8-1 Similarity in Right Triangles
Example 1b: Finding Geometric Means
Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.
6 and 15
Let x be the geometric mean.
Def. of geometric mean
Find the positive square root.
6
15
x
x
Cross multiply2 90x
3 10x
Holt Geometry
8-1 Similarity in Right Triangles
Example 1c: Finding Geometric Means
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.
Def. of geometric mean
Find the positive square root.
2
8
x
x
Cross multiply2 16x
4x
Holt 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 Geometry
8-1 Similarity in Right Triangles
Example: 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
Example 1
10
2
h
h
2 20h c
a b
nm
h
m = 2, n = 10, h = ___
c
a b
nm
h
c
a
b
n
m
hb
n
ha
m
h
2 10
20
2 5
h
h
2 10
Holt Geometry
8-1 Similarity in Right Triangles
Example 2
12
10
b
b
2 120b
c
a b
nm
h
m = 2, n = 10, b = ___
c
a b
nm
h
c
a
b
n
m
hb
n
ha
m
h
2 10
120
2 30
b
b
12
2 10
Holt Geometry
8-1 Similarity in Right Triangles
Example 2
30
27
b
b
2 810b
c
a b
nm
h
n = 27, c = 30, b = ___
c
a b
nm
h
c
a
b
n
m
hb
n
ha
m
h
27 30
810
9 10
b
b
27
Holt 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