7.1 Geometric Mean

Find the geometric mean between two
numbers

Solve problems involving relationships
between parts of right triangles and the
altitude to its hypotenuse

The geometric mean between two
numbers is the positive square root of their
products.

In other words, given two positive numbers
such as a and b, the geometric mean is the
positive number x such that
a:x=x:b
We can also write this as fractions,
a/x=x/b
or as cross products,
x 2 = ab.
Example 1a:
Find the geometric mean between 2 and 50.
Let x represent the geometric mean.
Definition of geometric mean
Cross products
Take the positive square
root of each side.
Simplify.
Answer: The geometric mean is 10.
Example 1b:
Find the geometric mean between 25 and 7.
Let x represent the geometric mean.
Definition of geometric mean
Cross products
Take the positive square
root of each side.
Simplify.
Use a calculator.
Answer: The geometric mean is about 13.2.
Your Turn:
a. Find the geometric mean between 3 and 12.
Answer: 6
b. Find the geometric mean between 4 and 20.
Answer: 8.9

If the altitude is
drawn to the
hypotenuse of a
right triangle, then
the two triangles
formed are similar to
the original triangle
and to each other.
∆CBD ~ ∆ABC
∆ACD ~ ∆ABC
∆CBD ~ ∆ACD

Theorem 7.2: In a right triangle, the
altitude from the right angle to the
hypotenuse divides the hypotenuse
into two segments. The length of the
altitude is the geometric mean of the
lengths of the two segments.

Theorem 7.3: In a right triangle, the
altitude from the right angle to the
hypotenuse divides the hypotenuse
into two segments. The length of
each leg of the right triangle is the
geometric mean of the lengths of the
hypotenuse and the segment of the
hypotenuse that is adjacent to the
leg.
BD
=
CD
CD
AD
AB
CB
=
CB
DB
AB
AC
=
AC
AD
6
x
=
18 = x2
√18 = x
x
3
5+2
y
7
y
=
y
=
2
y
2
√9 ∙ √2 = x
14 = y2
3 √2 = x
√14 = y
Example 2:
Example 2:
Cross products
Take the positive square
root of each side.
Use a calculator.
Answer: CD is about 12.7.
Your Turn:
Answer: about 8.5
Example 3:
KITES Ms. Turner is constructing a kite for her son.
She has to arrange perpendicularly two support rods,
the shorter of which is 27 inches long. If she has to
place the short rod 7.25 inches from one end of the
long rod in order to form two right triangles with the
kite fabric, what is the length of the long rod?
Example 3:
Draw a diagram of one of the right triangles formed.
Let
be the altitude drawn from the right angle of
Example 3:
Cross products
Divide each side by 7.25.
Answer: The length of the long rod is 7.25 + 25.2, or
about 32.4 inches long.
Your Turn:
AIRPLANES A jetliner has a wingspan, BD, of 211
feet. The segment drawn from the front of the plane to
the tail,
intersects
at point E. If AE is 163 feet,
what is the length of the aircraft?
Answer: about 231.3 ft
Example 4:
Find c and d in
Example 4:
is the altitude of right triangle JKL. Use Theorem 7.2
to write a proportion.
Cross products
Divide each side by 5.
Example 4:
is the leg of right triangle JKL. Use the Theorem 7.3 to
write a proportion.
Cross products
Take the square root.
Simplify.
Use a calculator.
Answer:
Your Turn:
Find e and f.
f
Answer:

Pre-AP Geometry
Pg. 346 #13 – 38 & #44

Geometry
Pg. 346 #13 – 32, #35 - 38