Name: ________________
Worksheet – Geometric Mean
Geometry
Feb. 24, 2012
I. Define the Geometric Mean:
If
a x
 then x is the geometric mean between a and b.
x b
-Example: What is the geometric mean between 4 and 9?
4 x

x 9
 x 2  36  x  6
Recall that the arithmetic mean between 4 and 9 is
49
 6 .5
2
Using the properties of proportions we can simplify the formula for the geometric mean
between a and b:
a x

x b
 x 2  a  b  x  ab
Practice: Find the geometric mean between the following pairs of numbers or
expressions:
a. 3, 48
(12)
b. 4, 18
(6√2)
c. 12, 30
d. 5𝑦, 30𝑦
(6√10)
II. Similarity in Right Triangles.
̅̅̅̅ from vertex 𝐶 to
1. Draw right ∆𝐴𝐵𝐶 with altitude 𝐶𝐷
the hypotenuse. Label the side lengths a, b, and c, the
altitude z, and the segments of the hypotenuse x and y,
as shown at right.
(5𝑦√6)
C
a
b
z
x
y
2. How many right triangles are there in the figure?
A
D
How many of the triangles are similar? Can you write a
c
similarity statement for the triangles in this figure?
- to make it easier to see the similar triangles and which parts are corresponding,
draw the three triangles separately with corresponding orientation, labeling all vertices
and sides in each triangle. See figures on the reverse side.
3. Write the similarity statement for these triangles:
B
C
4. From this similarity statement, derive some proportional relationships between a, b,
c, x, y, and z. Look for geometric means.
B
x z b
b


a. for triangles I & II:
From this we see that z is the
y a
2
geometric mean between x and y: z  xy
c b a
 
b. for triangles I & III:
From this we see that b
b x z
2
is the geometric mean between c and x: b  cx
c a b
 
c. for triangles II & III:
From this, we
a y z
z
z
I
D
B
c
a
III
A
see that a is the geometric mean between c and y:
x
A
b
C
y
a
II
a  cy
2
C
d. We can summarize the statements above using the familiar terms for the parts of
the right triangle:
-
The altitude is the geometric mean between the segments of the hypotenuse.
Each leg is the geometric mean between the near segment of the hypotenuse
and the whole hypotenuse.
e. Also note that
c b
 or c  z  b  a , which can be stated as: the product of the
a z
legs equals the product of the altitude and the hypotenuse.
Examples: use the geometric mean relationships just derived to determine the values of
the variables in the figures below:
18
x
7
b
w
h
y
z
6
3
a
ℎ2 = 3 ∙ 7 = 21 ⟹ ℎ = √21
62 = 𝑦 ∙ 18 ⟹ 𝑦 = 36 ÷ 18 = 2
𝑎2 = 3 ∙ 10 = 30 ⟹ ℎ = √30
𝑥 = 18 − 𝑦 = 18 − 2 = 16
𝑏 2 = 7 ∙ 10 = 70 ⟹ ℎ = √70
𝑤 2 = 16 ∙ 18 ⟹ 𝑤 = √16 ∙ 18 = 12√2
𝑧 2 = 2 ∙ 16 = 32 ⟹ 𝑧 = 4√2
z
D
Name: ________________
Homework Problems:
A. Simplify the following radicals:
1. √12
4. √
3. 3√54
2. √75
3
5.
5
√8
√6
6.
5
√15
B. Find the arithmetic and geometric means for each pair of values:
7. 2, 50
8. 14, 49
9. 6𝑥, 42𝑥
10. 22, 55
C. The figures below each show a right triangle with an altitude drawn to the
hypotenuse. Determine the values for the variables in each figure.
d = ______
e
f
d
e = ______
9
4
f = ______
g = ______
g
h = ______
h
j = ______
12
5
j
k = ______
k
16
m
12
m = ______
n = ______
25
p = ______
r
p
24
r = ______
n
t = ______
u
v
t
u = ______
v = ______
1
1
3
6
5
z
x
0.8
x = ______
y
y = ______
z = ______
a = ______
c
b = ______
b
12
c+7
c = ______
a
D. Use the figure at right and complete the following:
a
a.
a 2  ______
and
b
b 2  ______
d
b. Add the equations in part (a), factor the sum on
the right. Remember that 𝑐 = 𝑑 + 𝑒.
c. What famous theorem have you just proved?
e
c