8-1
8-1 Similarity
SimilarityininRight
RightTriangles
Triangles
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Geometry
Holt
Geometry
8-1 Similarity in Right Triangles
Warm Up
1. Write a similarity statement
comparing the two triangles.
∆ADB ~ ∆EDC
Simplify.
2.
3.
Solve each equation.
4.
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5. 2x2 = 50 ±5
8-1 Similarity in Right Triangles
Objectives
Use geometric mean to find segment
lengths in right triangles.
Apply similarity relationships in right
triangles to solve problems.
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8-1 Similarity in Right Triangles
Vocabulary
geometric mean
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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.
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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.
W
Z
By Theorem 8-1-1, ∆UVW ~ ∆UWZ ~ ∆WVZ.
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8-1 Similarity in Right Triangles
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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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
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, or x2 = ab.
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
x = 10
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Def. of geometric mean
Find the positive square root.
8-1 Similarity in Right Triangles
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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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
x=4
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Def. of geometric mean
Find the positive square root.
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.
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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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.
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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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.
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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)
x=4
6 is the geometric mean of
9 and x.
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.
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8-1 Similarity in Right Triangles
Helpful Hint
Once you’ve found the unknown side lengths,
you can use the Pythagorean Theorem to check
your answers.
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8-1 Similarity in Right Triangles
Check It Out! Example 3
Find u, v, and w.
92 = (3)(u)
u = 27
9 is the geometric mean of
u and 3.
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)
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v is the geometric mean of
u + 3 and 3.
Find the positive square root.
8-1 Similarity in Right Triangles
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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8-1 Similarity in Right Triangles
Example 4 Continued
Let x be the height of the tree above eye level.
(7.8)2 = 1.6x
7.8 is the geometric mean of
1.6 and x.
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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8-1 Similarity in Right Triangles
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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8-1 Similarity in Right Triangles
Check It Out! Example 4 Continued
Let x be the height of cliff above eye level.
(28)2 = 5.5x
x  142.5
28 is the geometric mean of
5.5 and x.
Divide both sides by 5.5.
The cliff is about 142.5 + 5.5, or
148 ft high.
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8-1 Similarity in Right Triangles
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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12
8-1 Similarity in Right Triangles
Lesson Quiz: Part II
For Items 3–6, use ∆RST.
3. Write a similarity statement comparing the
three triangles. ∆RST ~ ∆RPS ~ ∆SPT
4. If PS = 6 and PT = 9, find PR. 4
5. If TP = 24 and PR = 6, find RS.
6. Complete the equation (ST)2 = (TP + PR)(?). TP
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