SECTION 8.1
GEOMETRIC MEAN
When the means of a proportion are the same number, that number is
called the geometric mean of the extremes. The geometric mean
between two numbers is the positive square root of their product.
Example 1:
a) Find the geometric mean between 2 and 50.
2 x

x 50
Definition of geometric mean
x 2  100
Cross products
x  100
Take the positive square root of each side.
x = 10
Simplify.
b) Find the geometric mean between 3 and 12.
3 x

x 12
Definition of geometric mean
x 2  36
Cross products
x  36
Take the positive square root of each side.
x=6
Simplify.
Altitude of a triangle: The altitude of a triangle is a segment
drawn from a vertex to the line containing the opposite side and
perpendicular to the line containing that side.
Geometric Means in Right Triangles: In a right triangle, the
altitude drawn from the vertex of the right angle to the hypotenuse
forms two additional right triangles. These three right triangles
share a special relationship.
Example 2: Write a similarity statement identifying the three
similar triangles in the figure.
Separate the triangles into two triangles
along the altitude.
Then sketch the three triangles, reorienting the smaller ones so
that their corresponding angles and sides are in the same position
as the original triangle.
So, by Theorem 8.1, ΔEGF ~ ΔFGH ~ ΔEFH.
By definition of similar polygons, you can write proportions
comparing the side lengths of these triangles.
Notice that the circled relationships involve geometric means. This
leads to the next theorem.
*Pete always parachutes from the RIGHT ANGLE of the LARGE
triangle.
*The path he travels is the GEOMETRIC MEAN (so it is used
TWICE in the proportion).
*Then he visits TWO cities: (these are the other two blanks in the
proportion).
If the path he traveled was the MIDDLE path, then he visits the city
on the RIGHT or the city on the LEFT.
If the path he traveled was an OUTSIDE path, then he visits the
CLOSE city or the FAR city.
A
C
B
PB
BA
=
BC
PB
A
C
B
PA
AB
=
AC
PA
A
C
B
PC
CA
=
CA
PC
Example 3:
a) Find c, d, and e.
Since e is the measure of the altitude drawn to the hypotenuse of right ΔJKL,
e is the geometric mean of the lengths of the two segments that make up the
hypotenuse, JM and ML.
e  JM  ML
Geometric Mean (Altitude) Theorem
 6  24
Substitution
= 12
Simplify.
d  JM  JL
Geometric Mean (Leg) Theorem
 6   6  24 
Substitution
 180 = 6 5
Simplify.
c  ML  JL
Geometric Mean (Leg) Theorem
 24   24  6 
Substitution
 720 =12 5
Simplify.
Example 3:
b) Find e to the nearest tenth.
e  RU  RS
 16  16  4 
 320 =17.9
Geometric Mean (Leg) Theorem
Substitution
Simplify.
Example 4: Ms. Alspach is constructing a kite for her son. She has to
arrange two support rods so that they are perpendicular. The shorter rod is
27 inches long. If she has to place the short rod 7.25 inches from one end
of the long rod in order to form two right triangles with the kite fabric,
what is the length of the long rod?
Draw a diagram of one of the right triangles formed.
Let YX be the altitude drawn from the right angle of ΔWYZ.
YX  WX  XZ
Geometric Mean (Altitude) Theorem
13.5  7.25  XZ
Substitution
182.25  7.25  XZ
Square each side
25.14 ≈ XZ
Divide each side by 7.25
The length of the long rod is 7.25 + 25.14, or about 32.39 inches long.
Example 5: A jetliner has a wingspan, BD, of 211 feet. The segment
drawn from the front of the plane to the tail, AC intersects BD, at point
E. If AE is 163 feet, what is the length of the aircraft to the nearest tenth
of a foot?
Try it, on your own, the answer is 231.3 ft, did you get that?