Angle bisector 

Angle bisector theorem

Angle Bisector Theorem Postulate:

The angle bisector theorem states that the line that bisects an angle in a triangle will divide the opposite side in segments whose lengths are proportional to the lengths of the other two sides in the triangle.

Let us understand angle bisector theorem with the above figure:

Consider angle ABC, or simply angle B.

Let the line BD bisect angle B, i.e. line BD is angle bisector of angle B.

Again, let the angle bisector BD divide the opposite side AC into segments AD and DC, whose respective lengths are x and y.

Let the lengths of the other two sides AB and BC in the triangle ABC be c and arespectively. 

Now, according to the statement of the angle bisector theorem, the lengths of the two line segments AD and DC formed by the angle bisector BD will be proportional to the lengths of the other two sides AB and BC in the triangle.

i.e.

AD/DC = BA/BC, i.e.

x/y = c/a

In the following figure, the line segment AD bisects angle A and divides the opposite side, i.e BC into two parts BD and DC, having lengths a and b respectively.

Therefore, from the angle bisector theorem, we can write
AB/AC = a/b, i.e. p/q = a/b

Example:

The lengths of the three sides in a triangle are 4, 7 and 10. A line L divides the angle opposite which the side length is 10. Find the ratio of the lengths of the segments formed along the opposite side.

Solution:

Since line segment AD bisects angle A, therefore, from the angle bisector theorem, we can write

AB/AC = BD/DC= x/y

i.e. 4/7 = x/ (10 – x)     {since, y = 10 – x }

So, 40 – 4x = 7x, i.e.

11x = 40, so, x = 40/11