Triangle Angle Bisector Theorem

An angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to the other two sides of the triangle.

By the Angle Bisector Theorem,

BD DC = AB AC

Proof:

Draw BE AD .

Extend CA ¯ to meet BE at point E .

By the Side-Splitter Theorem,

CD DB = CA AE ---------( 1 )

The angles 4and1 are corresponding angles.

So, 41 .

Since AD ¯ is a angle bisector of the angle CAB,12 .

By the Alternate Interior Angle Theorem, 23 .

Therefore, by transitive property, 43 .

Since the angles 3and4 are congruent, the triangle ΔABE is an isosceles triangle with AE=AB .

Replacing AE by AB in equation ( 1 ),

CD DB = CA AB

Example:

Find the value of x .

By Triangle-Angle-Bisector Theorem,

AB BC = AD DC .

Substitute.

5 12 = 3.5x

Cross multiply.

5x=42

Divide both sides by 5 .

5x 5 = 42 5 x=8.4

The value of x is 8.4 .