CK-12 Geometry-Concepts

7.10. Proportions with Angle Bisectors http://www.ck12.org

By definition,

−→

ACdivides^6 BADequally, so^6 BAC∼=^6 CAD. The proportional relationship isCDBC=ADAB.

Theorem:If a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional
to the lengths of the other two sides.

Example A

Findx.

Because the ray is the angle bisector it splits the opposite side in the same ratio as the sides. So, the proportion is:

9

x

=

21

14

21 x= 126

x= 6

Example B

Determine the value ofxthat would make the proportion true.

You can set up this proportion just like the previous example.

5

3

=

4 x+ 1

15

75 = 3 ( 4 x+ 1 )

75 = 12 x+ 3

72 = 12 x

6 =x