CK-12 Geometry Concepts

http://www.ck12.org Chapter 8. Right Triangle Trigonometry

Geometric Mean Theorem #1:In a right triangle, the altitude drawn from the right angle to the hypotenuse divides
the hypotenuse into two segments. The length of the altitude is the geometric mean of these two segments. In other
words,BCAC=ACDC.

Geometric Mean Theorem #2:In a right triangle, the altitude drawn from the right angle to the hypotenuse divides
the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of
the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. In other words,BCAB=DBABandDCAD=ADDB.

Both of these theorems are proved using similar triangles.

Example A

Write the similarity statement for the triangles below.

Ifm^6 E= 30 ◦, thenm^6 I= 60 ◦andm^6 T RE= 60 ◦.m^6 IRT= 30 ◦because it is complementary to^6 T RE. Line up
the congruent angles in the similarity statement. 4 IRE∼4IT R∼4RT E

We can also use the side proportions to find the length of the altitude.

Example B

Find the value ofx.

First, let’s separate the triangles to find the corresponding sides.

Now we can set up a proportion.

shorter leg in 4 EDG

shorter leg in 4 DF G

=

hypotenuse in 4 EDG

hypotenuse in 4 DF G

6

x

=

10

8

48 = 10 x

4. 8 =x

Example C

Find the geometric mean of 24 and 36.

x=

√

24 · 36 =

√

12 · 2 · 12 · 3 = 12

√

6

Example D

Find the value ofx.

Using similar triangles, we have the proportion

shortest leg of smallest 4

shortest leg of middle 4

=

longer leg of smallest 4

longer leg of middle 4

9

x

=

x

27

x^2 = 243

x=

√

243 = 9

√

3