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.

b) Find the geometric mean between 3 and 12.

Definition of geometric mean

Cross products

Take the positive square root of each side.

x = 10 Simplify.

2

50

x

x

2 100x

100x

Definition of geometric mean

Cross products

Take the positive square root of each side.

x = 6 Simplify.

3

12

x

x

2 36x

36x

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. 

CB

A

=PBPBBABC

PA

AB=

AC

PA

CB

A

CB

A

=PCPCCACA

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.

Geometric Mean (Altitude) Theorem

Substitution

= 12 Simplify.

e JM ML

6 24

Geometric Mean (Leg) Theorem

Substitution

Simplify.

d JM JL

6 6 24

180 =6 5

Geometric Mean (Leg) Theorem

Substitution

Simplify.

c ML JL

24 24 6

720 =12 5

Example 3: b) Find e to the nearest tenth.

Geometric Mean (Leg) Theorem

Substitution

Simplify.

e RU RS

16 16 4

320 =17.9

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 be the altitude drawn from the right angle of ΔWYZ.YX

Geometric Mean (Altitude) Theorem

Substitution

Square each side

YX WX XZ

13.5 7.25 XZ

182.25 7.25 XZ

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, at point E. If AE is 163 feet, what is the length of the aircraft to the nearest tenth of a foot?

intersects ,AC BD

Try it, on your own, the answer is 231.3 ft, did you get that?