G.SRT.B.5 STUDENT NOTES & PRACTICE WS #1 – geometrycommoncore.com
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Geometric mean…..? Is that the same mean referred to when we say MEAN, MEDIAN AND MODE? NO!! That
mean is actually called the Arithmetic Mean. The Arithmetic mean of n numbers is the sum of values of the
numbers divided by n. The Arithmetic Mean determines the ‘average’ number of the group.
Geometry has its own mean and is much different than the Arithmetic Mean. The Geometric Mean of two
numbers, a and b, is the unique number x such that:
a x

x b
This is quite a special situation, where x is used as denominator in one ratio and as a numerator in the other
and yet BOTH RATIOS EQUAL EACH OTHER. Let’s look at an example of this special situation.
The geometric mean of 4 and 9 is 6 because
4 6

6 9
36 = 36.
Let’s find the geometric mean of a few different values.
What is the geometric mean
of 2 and 8?
2 x

x 8
x 16
2
x 4
Test out the value to see if it works
2 4

4 8
16 = 16
What is the geometric mean
of 6 and 24?
What is the geometric mean
of 5 and 15?
6 x

x 2 144 x  12
x 24
5 x x 2  75 x   75

x 15
x 5 3
Test out the value to see if it works
Test out the value to see if it works
6 12

12 24
144 = 144
5
5 3
5 3 15

75 = 75
NYTS (Now You Try Some)
1. What is the geometric mean
of 9 and 16?
2. What is the geometric mean
of 10 and 40?
3. What is the geometric mean
of 8 and 16?
So the question becomes…. What does this have to do with geometry? Great question!! This unique
proportion appears in a special type of triangle and it has a lot to do with similarity.
G.SRT.B.5 STUDENT NOTES & PRACTICE WS #1 – geometrycommoncore.com
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Given any right triangle, ABC.
B
Drop the altitude (the height) from the right angle (B) to the
opposite side AC . Let the altitude be called BD .
o
x
C
A
This forms three triangles.
Inner Left
ADB
Inner Right
BDC
Outer Whole
ABC
B
o
x
A
C
D
B
If we look carefully at ABC we see that mB = 90, mA = o and
mC = x, in other words o + x + 90 = 180.
x
Knowing that the 3rd ’s of a must be we can determine some of
the missing angle values in the diagram, mABD = x (180 - 90 - o)
and that mCBD = o (180 - 90 - x).
o
o
x
A
C
D
B
Notice that all three triangles are similar to each other by AA. It is
this unique relationship that will create three different geometric
mean relationships.
o
x
A
GEOMETRIC MEAN #1 -- Using the fact that ADB  ABC,
we can set up the proportion:
C
D
B
B
x
left 
AD AB


whole AB AC
AB 2  AD( AC )
GEOMETRIC MEAN #2 -- Using the
fact that BDC  ABC, we can set
up the proportion:
right 
DC BC


whole BC
AC
o
A
o
D
x
A
C
B
B
o
x
o
D
C
x
A
C
BC 2  DC ( AC )
GEOMETRIC MEAN #3 -- Using the fact that ADB  BDC, we
can set up the proportion:
B
B
o
x
right  AD BD


left 
BD DC
BD2  AD( DC )
x
o
A
D
D
C
G.SRT.B.5 STUDENT NOTES & PRACTICE WS #1 – geometrycommoncore.com
The three geometric means are:
3
B
AB 2  AD( AC ) BC 2  DC ( AC ) BD2  AD( DC )
o
AB  AD( AC )
BC  DC ( AC )
A
BD  AD( DC )
x
D
C
Let’s apply these three geometric mean relationships to solve problems.
x
z
y
z
y
x
9 cm
12 cm
6 cm
z
y
x
5 cm
x
y 2  (4)(9)
x 2  (4)(13)  52
z 2  (9)(13)  117
z 2  36
x   2 13  2 13
z   3 13  3 13
z  6  6
x 2  (12)(12  9)
z 2  (9)(9  12)
y 2  (12)(9)
x 2  (12)(21)  252
z 2  (9)(21)  189
z 2  108
x   6 7 6 7
z   3 21  3 21
z  6 3  6 3
From this point forward I will only list the positive value
that applies to the given situation.
62  ( x)(10)
36  10 x
x  3.6
z 2  (6.4)(10)
y 2  (3.6)(6.4)
z 2  64
z 2  23.04
z  8  8
z   4.8  4.8
52  (12)( x)
25  12 x
x  2.08
y 2  (2.08)(14.08)
z 2  (12)(14.08)
y 2  29.29
z 2  168.96
y   5.41  5.41
z   13.00  13.00
10 cm
z
y
z 2  (9)(4  9)
Only the positive value makes sense here
because we are solve for lengths.
9 cm
4 cm
x 2  (4)(4  9)
12 cm
Answers:
1. +12 or -12
2. +20 or -20
3. +8sqrt(2) or – 8sqrt(2)
4. x = 6sqrt(5) cm, y = 12 cm, z = 12sqrt(5) cm
5. x = 8sqrt(2) cm, y = 8 cm, z = 8sqrt(2)cm
6. x = 24 cm, z = 6sqrt(2) cm, y = 18sqrt(2) cm
6.
y
z
9 cm
3 cm
x
5.
8 cm
8 cm
x y
z
4.
24 cm
6 cm
y
x
z
NYTS (Now You Try Some)
Determine the values for x, y and z.
G.SRT.B.5 STUDENT NOTES & PRACTICE WS #1 – geometrycommoncore.com
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