Mean Proportional
Means and Extremes
In a proportion, the product of the
means is equal to the product of the
extremes

Means
a
b

c
d
Extremes
Mean Proportional
If the two means of the proportional are equal , it is called the
mean proportional between the first and fourth terms.
In the proportion
2
8

8
the mean proportional is?
32
8
The mean proportional is also known as the geometric mean
Page 38
Mean Proportional
2. Which of the following represents the geometric mean of 4 and 9?
(1) 6
(3) 18
(2) 6.5
(4) 36
4 𝑥
=
𝑥 9
𝑥 2 = 36
𝑥 = ±6
𝑥=6
𝑥 2 = 36
3. 8 is the geometric mean between 4 and which of the following values?
(1) 64
(3) 16
(2) 24
(4) 4
4 8
=
8 𝑥
4𝑥 64
=
4
4
4𝑥 = 64
𝑥 = 16
Page 39
Theorems
Right Triangle Altitude Theorem – If the altitude of a right triangle is
drawn (to the hypotenuse), then it divides the triangle into two similar
triangles that are similar to the original.
A
C
 DAC
 DBA
 ABC
D
 ABC ~  DBA ~  DAC
B
Page 39
Theorems
Mean proportionality theorem – The altitude of a right triangle is the
mean proportional between the segments into which it divides the
hypotenuse.
Start by filling in
the mean
Mean
Altitude
Propotionalproportional
A
Lets try with
some values
27
99

9
x
27 x  81
27
9
x 3
Solve for x!
C
x
D
Two parts of the
hypotenuse
27
27
B
Page 39
Notice that we did not have the ± in
front of our answer this time. Why?
6 𝑥
=
𝑥 3
𝑥 2 = 18
𝑥 2 = 18
𝑥 = 9∙2
𝑥=3 2
4 𝑥
=
𝑥 9
𝑥 2 = 36
𝑥 2 = 36
𝑥=6
Page 40
18
𝑥+9
=
𝑥+9
8
𝑥 + 9 𝑥 + 9 = 144
𝑥 2 + 9𝑥 + 9𝑥 + 81 = 144
𝑥 2 + 18𝑥 + 81 = 144
−144 − 144
𝑥 2 + 18𝑥 − 63 = 0
𝑥 + 21 𝑥 − 3 = 0
𝑥 + 21 = 0
𝑥−3=0
𝑥 = −21
𝑥=3
Page 43
9
12
=
12
𝑥
9𝑥 = 144
12
9
𝑥
9𝑥 144
=
9
9
𝑥 = 16
Page 43
4 8
=
8 𝑥
4𝑥 = 64
8
4
𝑥
4𝑥 64
=
4
4
𝑥 = 16
Homework
Page
43
#11,15,17,21,22,23
Separate Sheet
Page 43
3 6
=
6 𝑥
6
3𝑥 = 36
3
𝑥
3𝑥 36
=
3
3
𝑥 = 12
Page 43
4 𝑥
=
𝑥 9
𝑥
4
𝑥 2 = 36
9
𝑥 2 = 36
𝑥=6
Page 43
3
𝑥
=
𝑥 27
𝑥
𝑥 2 = 81
𝑥 2 = 81
3
27
𝑥=9
Page 43
5
𝑥
=
𝑥 20
𝑥 2 = 100
𝑥
𝑥 2 = 100
𝑥 = 10
5
20
Page 43
2 8
=
8 𝑥
2𝑥 = 64
8
2𝑥 64
=
2
2
2
𝑥
𝑥 = 32
Page 43
𝑥
6
=
6 9𝑥
9𝑥 2 = 36
6
9𝑥 2 36
=
9
9
𝑥
𝑥=2
9𝑥
9𝑥 = 2 ∙ 9 = 18
𝑥2 = 4
𝑥2 = 4
𝑥=2