Similarity in Right Triangles

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Similarity in Right Triangles
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry
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Warm Up 1. Write a similarity statement comparin
g the two triangles. Simplify. 2. 3. Solve
each equation. 4. 5. 2x2 50
?ADB ?EDC
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Objectives
Use geometric mean to find segment lengths in
right triangles. Apply similarity relationships
in right triangles to solve problems.
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Vocabulary
geometric mean
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In a right triangle, an altitude drawn from the
vertex of the right angle to the hypotenuse forms
two right triangles.
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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.
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Check It Out! Example 1
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, ?LJK ?JMK ?LMJ.
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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.
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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.
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Example 2B Finding Geometric Means
Find the geometric mean of each pair of numbers.
If necessary, give the answer in simplest radical
form.
5 and 30
Let x be the geometric mean.
x2 (5)(30) 150
Def. of geometric mean
Find the positive square root.
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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.
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Check It Out! Example 2b
Find the geometric mean of each pair of numbers.
If necessary, give the answer in simplest radical
form.
10 and 30
Let x be the geometric mean.
x2 (10)(30) 300
Def. of geometric mean
Find the positive square root.
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Check It Out! Example 2c
Find the geometric mean of each pair of numbers.
If necessary, give the answer in simplest radical
form.
8 and 9
Let x be the geometric mean.
x2 (8)(9) 72
Def. of geometric mean
Find the positive square root.
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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.
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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.
y is the geometric mean of 4 and 13.
y2 (4)(13) 52
Find the positive square root.
z2 (9)(13) 117
z is the geometric mean of 9 and 13.
Find the positive square root.
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Check It Out! Example 3
Find u, v, and w.
92 (3)(u) 9 is the geometric mean of u
and 3.
u 27 Divide both sides by 3.
w2 (27 3)(27) w is the geometric mean of
u 3 and 27.
Find the positive square root.
v2 (27 3)(3) v is the geometric mean of
u 3 and 3.
Find the positive square root.
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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?
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Example 4 Continued
Let x be the height of the tree above eye level.
7.8 is the geometric mean of 1.6 and x.
(7.8)2 1.6x
x 38.025 38
Solve for x and round.
The tree is about 38 1.6 39.6, or 40 m tall.
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Check It Out! 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?
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Check It Out! Example 4 Continued
Let x be the height of cliff above eye level.
(28)2 5.5x
28 is the geometric mean of 5.5 and x.
x ? 142.5
Divide both sides by 5.5.
The cliff is about 142.5 5.5, or 148 ft high.
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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
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Lesson Quiz Part II
For Items 36, use ?RST. 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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TP