Download Notes Geometric Mean 2015

NOTES GEOMETRIC MEAN /
SIMILARITY IN RIGHT TRIANGLES
I can use relationships in similar right triangles.
Simplifying Radicals
Perfect Squares – 1, 4, 9, 16, 25, 36, 49, 64, 81…
 Find the largest Perfect Square that goes into the
number evenly
example: 72
The largest Perfect Square that goes into 72 is 36.
= 36 x 2
=
x 2
=6 2

What if you picked 9 instead of 36?

If you pick a smaller Perfect Square you must reduce
more than once.
example:
72 9 is a Perfect Square that goes into 72 evenly, though not the largest
=
=
=
=
=
=
9x8
9x 8
3 8 8 can be divided by another Perfect Square, 4
3 4x2
3x2 2
6 2
Geometric Mean
Geometric Mean is the square root of the product of two values.
If a, b, and x are positive numbers and
geometric mean between a and b.
a x

x b
, then x is called the
Example : Find the geometric mean of 3 and 12.
3
= x
x
12
Write a proportion.
x2 = 36
Cross-Product Property
x2 =
36 Find the positive square root.
x=6
The geometric mean of 3 and 12 is 6.
Similarity in Right Triangles
Altitude – segment drawn from 90 degrees to the opposite side
Right Triangle Similarity 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 each other.
Similarity in Right Triangles - Corollary 1
alt
seg1
seg2
The length of the altitude of the right triangle is the geometric
mean between the segments of the hypotenuse .
seg1 alt

alt
seg 2
Example
Find the length of the altitude.
3
x
X
3
6
seg1 alt

alt
seg 2
=
18 = x2
√18 = x
√9 ∙ √2 = x
3 √2 = x
x
6
Similarity in Right Triangles – Corollary 2
leg
SHAL
hypotenuse
Each leg of the right triangle is the geometric mean between the
hypotenuse and the segment of the hypotenuse adjacent to the leg.
hypotenuse
leg

leg
SHAL
Example 3
2
y
Find the length of the leg.
5
hypotenuse
leg

leg
SHAL
y
5+2 =
y
2
7
y
=
y
2
14 = y2
√14 = y
Similarity in Right Triangles
Solve for x.
2
= 6
6
x
2x = 36
seg1 alt

alt
seg 2
hypotenuse
leg

leg
SHAL
x = 18
Write a proportion.
Cross-Product Property
Similarity in Right Triangles
Solve for y.
x
y
=
y
2+x
Write a proportion.
y
18
= 2 + 18
y
Substitute 18 for x.
y2 = 360
y=
y=6
360
10
Cross-Product Property.
Find the positive square root.
Write in the simplest radical form.