Download Geometric Mean

Geometry
8.1
Right Triangles
A radical is in
simplest form when:
1. No perfect square factor other than 1
is under the radical sign.
2. No fraction is under the radical sign.
3. No radical is in a denominator.
Simplify:
1. 81
2.
24
3. 300
5
4.
3
5.
250
48
6. 4 27
3. 10 3
15
4.
3
5 30
5.
12
6. 12 3
Answers:
1. 9
2. 2 6
Geometric Mean
If a, b and x are positive numbers and
mean
a
x
=
b
x
mean
then x is called the geometric mean between a and b.
Multiplying means/extremes we find that:
Taking the square root of each side:
x = ab
x = ab
2
A positive number
Geometric Mean
• Basically, to find the geometric mean of 2
numbers, multiply them and take the square
root.
• Note that the geometric mean always falls
between the 2 numbers.
Ex: Find the geometric mean between 5 and 11.
Use the formula
x = ab

5 11 =
55
Geometric Mean
x = ab
Ex: Find the geometric mean between the two numbers.
a. 5 and 20
5 20 =
100 = 10
Avoid multiplying large numbers together. Break
numbers into perfect square factors to simplify.
24 32 = 24
b. 24 and 32
=
4 6
32
2 16 = 2 6 4 2
= 8 12 = 8 4 3 = 16 3
Geometric Mean
x = ab
Find the geometric mean between the two numbers.
7. 5 and 20
5 20 =
100 = 10
8. 64 and 49
64 49 = 8 7 = 56
9. 1 and 3
1 3= 3
10.100 and 6
100 6 = 10 6
11.20 and 24
20 24 = 2 5 2 6 = 4 30
∆ Review
Hypotenuse the side opposite the right
angle in a right triangle
• Altitude the
perpendicular segment
from a vertex to the line
containing the opposite
side
Theorem
If the altitude is drawn to the hypotenuse of a right
triangle, then the two triangles formed are similar to the
original triangle and to each other.
~
a
b
a
~
b
Corollaries
When the altitude is drawn to the hypotenuse of a
right triangle:
Y
the length of the altitude
is the geometric mean
between the segments of
the hypotenuse.
X
Each leg is the geometric mean between the
hypotenuse and the segment of the
hypotenuse that is adjacent to that leg.
A
Z
Corollary 1
piece of hypotenuse
altitude
altitude
= other piece
of hypotenuse
X
XA YA
=
YA AZ
Y
A
Z
Corollary 2
hypotenuse
leg
=
leg
piece of
hyp. adj. to leg
Y
X
A
XZ XY
For leg XY :
=
XY XA
Z
Corollary 2
hypotenuse
leg
=
leg
piece of
hyp. adj. to leg
Y
X
A
XZ YZ
For leg YZ :
=
YZ AZ
Z
Exercises
C
8
A
x
12. If CN = 8 and NB = 16, find AN. Let x = AN
long leg
short leg
x
8

8 16
8  16 x 64  16 x
2
x4
N 16 B
Exercises
C
x
A
8
14. If AN = 8 and NB = 12, find CN. Let x = CN
long leg
short leg
8
x

x 12
x  96 x  96  6 16
2
x4 6
N 12 B
Exercises
C
12
N x
18
17. If AB = 18 and CB = 12, find NB. Let x = NB
A
hypot.
short leg
18 12

12
x
12  18 x 144  18 x
2
x8
B
Answers
Exercises 13 - 19
13.
15.
16.
18.
19.
36
20
2√15
25
5
Homework
pg. 288 #16-30, 31-39 odd