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
4π₯
=π₯9
π₯2=36
βπ₯2=β36
π₯=Β±6
π₯=6
48=
8π₯
4 π₯=64 π₯=16
4 π₯4
=644
Theorems
Page 39
Right Triangle Altitude Theorem β 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 β 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
6π₯
=π₯3
π₯2=18
βπ₯2=β18
π₯=β9 β2
π₯=3β2
Notice that we did not have the in front of our answer this time. Why?
4π₯
=π₯9
π₯2=36
βπ₯2=β36π₯=6
Page 40
18π₯+9
=π₯+9
8
(π₯+9 ) (π₯+9 )=144
π₯2+9π₯+9 π₯+81=144
π₯2+18 π₯+81=144β144β144
π₯2+18 π₯β63=0
(π₯+21 ) (π₯β3 )=0π₯+21=0π₯=β21
π₯β3=0π₯=3
REJECT
Page 43
1 2
9 π₯
912
=12π₯
9 π₯=1449π₯9
=144
9
π₯=16
Page 43
8
4 π₯
48=
8π₯
4 π₯=644 π₯4
=644
π₯=16
Homework
Page 43#11,15,17,21,22,23
Separate Sheet
Page 43
6
3 π₯
36=
6π₯
3 π₯=363π₯3
=363
π₯=12
Page 43
94
π₯
4π₯
=π₯9
π₯2=36
βπ₯2=β36π₯=6
Page 43
273
π₯
3π₯
=π₯
27
π₯2=81
βπ₯2=β81π₯=9
Page 43
205
π₯
5π₯
=π₯
20
π₯2=100
βπ₯2=β100π₯=10
Page 43
π₯2
8
28=
8π₯
2 π₯=642π₯2
=642
π₯=32
Page 43
9 π₯π₯
6
π₯6=
69 π₯
9 π₯2=36
βπ₯2=β4
π₯2=4
π₯=2
9π₯2
9=36
9
π₯=2 9 π₯=2 β9=18
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