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Similarity in Right Triangles
Theorem 7-3: The altitude to the hypotenuse of a right triangle
divides the triangles into two triangles that are similar to the original
triangle and to each other.
Geometric Mean
Review: How do we find the arithmetic mean of 3 and 27?
a
x
Geometric Mean: The number x such that  , where a, b, and x
x b
are positive numbers
The geometric mean, in mathematics, is a type of mean or average, which indicates the
central tendency or typical value of a set of numbers.

Find the geometric mean of 3
and 27.
Find the geometric mean of 4
and 18.
Note: x  ab
Purpose of the Geometric Mean
1. The geometric mean can give a meaningful "average"
to compare two companies.
2. The use of a geometric mean "normalizes" the ranges
being averaged, so that no range dominates the
weighting.
3. The geometric mean applies only to positive
numbers.[2]
4. It is also often used for a set of numbers whose values
are meant to be multiplied together or are exponential
in nature, such as data on the growth of the human
population or interest rates of a financial investment.
Geometric Mean
6.75in
5.2 in
8.75in
Corollary to Theorem 7-3: The length of the altitude to the
hypotenuse of a right triangle is the geometric mean of the lengths
of the segments of the hypotenuse
Similarity in Right Triangles
Find the values of x and y in the following right triangle.
X
Y
Y
4
5
X
4
+
5
You Try One!!!
Find the values of x and y in the following right triangle.
Proof of Corollary to Theorem 7-3
C
Given : Right triangle, ABC, with
CD the altitude to the hypotenuse
AD CD
Prove :

CD DB
A
Statements
1. Right triangle, ABC, with
D
B
Reasons
1.
CD the altitude to the hypotenuse
2.
AD CD

3. CD DB
2. Altitude of rt. Δ to hypotenuse
divides into 2 ~ Δs
3.
Real World Connection
As Marla arrives at the lake from the parking lot, she
reads a sign that says she is 320m from the dock. How
far is Marla from the information center?
Kick it up a notch!
Find the value of x in the following right triangle.
2x - 1
1
x
Similarity in Right Triangles
m1  m4  m7
m2  m6  m8
m3  m5  m9