The student can use the geometric mean to find segment lengths in right triangles and apply similarity relationships to solve problems. Specifically: - The altitude to the hypotenuse of a right triangle forms two triangles similar to each other and the original triangle. Proportions can be set up between corresponding sides. - The length of the altitude to the hypotenuse is the geometric mean of the lengths of the two segments it forms. - The length of a leg is the geometric mean of the lengths of the hypotenuse and the adjacent segment. Setting up similar triangles allows solving for unknown side lengths.

1. 4.11.3 Similar Right Triangles
The student is able to (I can):
• Use geometric mean to find segment lengths in right
triangles
• Apply similarity relationships in right triangles to solve
problems.

2. geometric
mean
A proportion with the following pattern:
or or
Example: Find the geometric mean of 5 and
20.
=
a x
x b
=2
x ab =x ab
=
5 x
x 20
=2
x 100
x = 10

3. Theorem The altitude to the hypotenuse of a right
triangle forms two triangles that are
similar to each other and to the original
triangle.
T
I
ME
I T
M
E
I
T
IE
M
90˚−∠M
90˚−∠M
∆MIT ~ ∆IET ~ ∆MEI

4. Because the triangles are similar, we can
set up proportions between the sides:
T
I
ME
I T
M
E
I
T
IE
M
∆MIT ~ ∆IET ~ ∆MEI
= =
MI IE ME
, etc.
IT ET EI

5. Corollary The length of the altitude to the
hypotenuse is the geometric mean of the
lengths of the two segments.
H
E
AT
ab
xy
c
h
=
x h
h y
or =2
h xy

6. The length of a leg of a right triangle is the
geometric mean of the lengths of the
hypotenuse and the segment of the
hypotenuse adjacent to that leg.
H
E
AT
ab
xy
c
h
=
x a
a c
or =2
a xc
=
y b
b c
or =2
b yc

7. When you are setting up these problems,
remember you are basically setting up
similar triangles.
Example
1. Find x, y, and z.
9 6
x
y
z
=
9 6
6 x
9x = 36
x = 4
=
+
9 z
z 9 4
=2
z 117
=z 3 13
=
+
4 y
y 9 4
=2
y 52
=y 2 13