mean proportional. means and extremes means extremes in a proportion, the product of the means is...
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Mean Proportional
Means and Extremes
d
c
b
a
Means
Extremes
In a proportion, the product of the means is equal to the product of the extremes a
dc
b
Mean Proportional
If the two means of the proportional are equal , it is called the mean proportional between the first and fourth terms.
32
8
8
2In the proportion the mean proportional is?
8The mean proportional is also known as the geometric mean
Mean Proportional
2. Which of the following represents the geometric mean of 4 and 9?
(1) 6 (3) 18
(2) 6.5 (4) 36
Page 38
3. 8 is the geometric mean between 4 and which of the following values?
(1) 64 (3) 16
(2) 24 (4) 4
4𝑥
=𝑥9
𝑥2=36
√𝑥2=√36
𝑥=±6
𝑥=6
48=
8𝑥
4 𝑥=64 𝑥=16
4 𝑥4
=644
Theorems
Page 39
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 BD
DACDBAABC ~~
ABCDBA
DAC
Theorems
Page 39
Mean proportionality theorem – The altitude of a right triangle is the mean proportional between the segments into which it divides the hypotenuse.
A
C BD
AltitudeMean Propotional
Two parts of the hypotenuse
Lets try with some values
Solve for x!9
x 27
9
9x
9
9
27
8127 x27 27
3x
Start by filling in the mean
proportional
Page 39
6𝑥
=𝑥3
𝑥2=18
√𝑥2=√18
𝑥=√9 ∙2
𝑥=3√2
Notice that we did not have the in front of our answer this time. Why?
4𝑥
=𝑥9
𝑥2=36
√𝑥2=√36𝑥=6
Page 40
18𝑥+9
=𝑥+9
8
(𝑥+9 ) (𝑥+9 )=144
𝑥2+9𝑥+9 𝑥+81=144
𝑥2+18 𝑥+81=144−144−144
𝑥2+18 𝑥−63=0
(𝑥+21 ) (𝑥−3 )=0𝑥+21=0𝑥=−21
𝑥−3=0𝑥=3
REJECT
Page 43
1 2
9 𝑥
912
=12𝑥
9 𝑥=1449𝑥9
=144
9
𝑥=16
Page 43
8
4 𝑥
48=
8𝑥
4 𝑥=644 𝑥4
=644
𝑥=16
Homework
Page 43#11,15,17,21,22,23
Separate Sheet
Page 43
6
3 𝑥
36=
6𝑥
3 𝑥=363𝑥3
=363
𝑥=12
Page 43
94
𝑥
4𝑥
=𝑥9
𝑥2=36
√𝑥2=√36𝑥=6
Page 43
273
𝑥
3𝑥
=𝑥
27
𝑥2=81
√𝑥2=√81𝑥=9
Page 43
205
𝑥
5𝑥
=𝑥
20
𝑥2=100
√𝑥2=√100𝑥=10
Page 43
𝑥2
8
28=
8𝑥
2 𝑥=642𝑥2
=642
𝑥=32
Page 43
9 𝑥𝑥
6
𝑥6=
69 𝑥
9 𝑥2=36
√𝑥2=√4
𝑥2=4
𝑥=2
9𝑥2
9=36
9
𝑥=2 9 𝑥=2 ∙9=18