Warm Up
1. Write a similarity statement
comparing the two triangles.
∆ADB ~ ∆EDC
Estimate to the nearest integer.
2.
3.
Solve each equation.
4.
5. 2x2 = 50
±5
10-3 Similarity in Right
Triangles
Common Misconceptions
 Similarity vs Congruence
 Congruent or Similar?
 The two shapes need to be the same size to be
congruent.
 For example, these are congruent regular pentagons
 When we need to resize one shape to make it the
same as the other, the shapes are Similar
 Here are two similar regular pentagons
Theorem 1
In a right triangle, an altitude drawn from the vertex of the
right angle to the hypotenuse forms two right triangles.
Theorem 10-3
Example 1- Identifying Similar
Right Triangle
Write a similarity statement comparing the
three triangles.
Sketch the three right triangles with the angles of the triangles
in corresponding positions.
By Theorem 10-3, ∆UVW ~ ∆UWZ ~ ∆WVZ.
Example 2-On your own
Write a similarity statement comparing the
three triangles.
Sketch the three right triangles with the angles of
the triangles in corresponding positions.
By Theorem 10-3, ∆LJK ~ ∆JMK ~ ∆LMJ.
Definition 1
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.
Warm Up
Example 3: 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
Def. of geometric mean
Find the positive square root.
Example 4: On your own!
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.
You can use Theorem 10-3 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.
Corollaries 1 & 2
Corollary 1
Corollary 2
Example 5: Finding side
lengths in right triangles
Find x, y, and z.
h2 = (x)(y)
62 = (x)(9)
x=4
6 is the geometric mean of 9 and x.
Divide both sides by 9.
a2 = (x)(c)
y2 = (4)(13) = 52
y is the geometric mean of 4 and 13.
Find the positive square root.
b2 = (y)(c)
z2 = (9)(13) = 117
z is the geometric mean of 9 and 13.
Find the positive square root.
REMEMBER!!!
Helpful Hint
Once you’ve found the unknown side lengths, you can use the Pythagorean
Theorem to check your answers.
Example 6: On your own!
Find u, v, and w.
h2 = (x)(y)
92 = (3)(u)
u = 27
9 is the geometric mean of u and 3.
Divide both sides by 3.
b2 = (y)(c)
w2 = (27 + 3)(27)
w is the geometric mean of u + 3 and 27.
Find the positive square root.
a2 = (x)(c)
v2 = (27 + 3)(3)
v is the geometric mean of u + 3 and 3.
Find the positive square root.
Example 7
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?
Let x be the height of the tree
above eye level.
h2 = (x)(y)
(7.8)2 = 1.6y
y = 38.025 ≈ 38
7.8 is the geometric mean of 1.6 and y.
Solve for y and round.
The tree is about 38 + 1.6 = 39.6, or 40 m tall.
Example 8-on your own!
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?
Let x be the height of cliff above eye level.
(28)2 = 5.5x
y  142.5
28 is the geometric mean
of 5.5 and y.
Divide both sides by 5.5.
The cliff is about 142.5 + 5.5, or 148 ft high.
Homework
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