Geometric
Mean,
Made Easy!
Geometric Mean With Two
Numbers
The geometric mean of two numbers is the square root of their
product. When asked to find the geometric mean with two
numbers, use the proportion below. Always leave x as your
variable. Plug your numbers into a and b. Then, cross multiply
and solve.
a
x
x = b
Geometric Mean With Two
Numbers
Watch Me!
Find the geometric mean between 3 and 12.
3 = x
x
12
x·x = 3·12
2
x
√ = √ 36
x=6
Your Turn!
Find the geometric mean between 5 and 45.
5 = x
x
45
x·x = 5·45
√x 2 = √ 225
x = 15
Geometric Mean With Triangles
Method 1 aka “The Altitude Method”:
There is a variable or number on the altitude of the BIG
right triangle.
18 = x
x
8
x
18
2
x·x = 18·8
2
x
√ = √ 144
x = 12
Geometric Mean With Triangles
1.  Circle the variable on the leg you are solving for
Method
aka to
“The
Leg Method”:
2.  Transfer
your2circle
the angle
opposite of your
variable
in theor
BIG
right triangle
There is
a variable
number
on one or both legs of the
3.  Start from thatBIG
angle,
always
traveling on the
right
triangle.
hypotenuse of the BIG right triangle FIRST
x
5
y
2
7 = x
x
5
x·x = 7·5
2
x
√ = √ 35
x = 5.92
Geometric Mean With Triangles
1.  Circle the variable on the leg you are solving for
Method
aka to
“The
Leg Method”:
2.  Transfer
your2circle
the angle
opposite of your
variable
in theor
BIG
right triangle
There is
a variable
number
on one or both legs of the
3.  Start from thatBIG
angle,
always
traveling on the
right
triangle.
hypotenuse of the BIG right triangle FIRST
x
5
y
2
7 = y
y
2
y·y = 7·2
2
y
√ = √ 14
y = 3.74
Geometric Mean With Triangles
SomeSo,
triangles
have
variables
or
numbers onxaltitude
AND
let’s
start
with
x…since
is
on
legs of the BIG right triangle. In that case, use both
methods.
for the variable
comes
theSolve
altitude,
we willthatuse
ourfirst in the
alphabet, then so on…
Altitude Method!
y
z
x
9
3
9 = x
x
3
x·x = 9·3
2
x
√ = √ 18
x = 4.24
Geometric Mean With Triangles
Now, y…since y is on one of the
legs, we will use our Leg Method!
y
9
x
z
3
12 = y
y
9
y·y = 12·9
2
y
√ = √ 108
y = 10.39
Geometric Mean With Triangles
Now, z…since z is on one of the
legs, we will use our Leg Method!
y
9
x
z
3
12 = z
z
3
z·z
2
z
√ z
= 12·3
= √ 36
=6
Check This One Out…
This problem looks a little different. The x is not on the altitude or one
of the legs. But, remember we have only two methods to choose from.
What method do we have information for? Altitude or Leg Method?
Since there is NO information on the altitude whatsoever, we will use
our Leg Method! Start with the information given on your leg…
15
x = 15
15
5
15·15 = 5·x
225 = 5x
5
x
= 45
[-------------------x-------------------]
Check This One Out…
This problem, too, looks a little different. The x is not on the altitude
or one of the legs. But, remember we have only two methods to
choose from. What method do we have information for this time?
Altitude or Leg Method?
Since there IS information on the altitude, we will use our Altitude
Method! Start from the corner…
x = 4
4
6
4
x
6
4·4 = 6·x
16 = 6x
x = 2.7