Geometry
Lesson 8 – 1
Geometric Mean
Objective:
Find the geometric mean between two numbers.
Solve problems involving relationships between parts of a right triangle
and the altitude to its hypotenuse.
Geometric Mean
Geometric mean

The positive square root of their product.
a x
2
 , so x  ab and x  ab
x b
Geometric mean between 9 and 4.
9 x
2
 , so x  36 and x  36
x 4
6
Find the geometric
mean between 8 & 10.
x  ab
x  8  10
Simplify the radical!
x  24 25
What is the square root of 16?
x4 5
Find the geometric mean
5 & 45
x  ab
x  5 45
x  559
Square root of 25?
Square root of 9?
x = 5(3)
x = 15
12 & 15
x  ab
x  12 15
x  3 4  3 5
x  3(2) 5
x6 5
Theorem
Theorem 8.1

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.
Write a similarity statement
identifying the three similar right
triangles in the figure.
The three triangles:
FJG ~
GJH ~
FGH
Write a similarity statement
identifying the three similar right
triangles in the figure.
KML ~
MPL ~
KPM
STR ~
QTS ~
QSR
Theorem
Geometric Mean (Altitude) Theorem

The altitude drawn to the hypotenuse of a
right triangle separates the hypotenuse into
two segments
Theorem
Geometric Mean (Leg) Theorem

The altitude drawn to the hypotenuse of a right triangle
separates the hypotenuse into two segments. The length
of a leg of this triangle is the geometric mean between
the the length of the the hypotenuse and the segment of
the hypotenuse adjacent to that leg.
Find x, y, and z.
20 x
25 y


x 5
y 5
x  100
y  125
x = 10
25 z (leg theorem)

z 20
z  500  10 5  22.4
 5 5  11.2
Find x, y, & z
8 z

z 25
z  200
z  10 2  14.1
33 y

y 25
y  825
y  5 33  28.7
33 x

x 8
x  264
x  2 66  16.2
Find x, y, and z.
x 12

12 9
9x = 144
x = 16
16
25 z

z 9
25 y

y 16
z  225
y  400
z = 15
y = 20
Zach wants to order a banner that will hang over the
side of his high school baseball stadium grandstand
and reach the ground. To find this height, he uses a
cardboard square to line up the top and bottom of the
grandstand. He measures his distance from the
grandstand and from the ground to his eye level.
Find the height of the grandstand to the nearest foot.
5.75 10.5

10.5
x
5.75x = 110.25
x  19.17
Grandstand = 19.17 + 5
The grandstand is about 25 feet tall.
Homework
Pg. 535 1 – 7 all, 8 – 24 E,
28 – 36 EOE, 50, 54 – 74 EOE