Question

Given that is the angle bisector of

Answer 1

Notice that AD also creates two supplementary angles, BDA and ADE. Let y be the measure of the smaller angle, ADE; then the measure of the larger angle, BDA, is 180° - y.

Since AD bisects angle A, we have by the law of sines

sin(A/2)/x = sin(y)/8

and

sin(A/2)/6 = sin(180° - y)/12

Now,

sin(180° - y) = sin(180°) cos(y) - cos(180°) sin(y) = sin(y)

so that

sin(A/2) = x/8 sin(y)

and

sin(A/2) = 6/12 sin(y) = 1/2 sin(y)

Solve for x :

x/8 sin(y) = 1/2 sin(y)

x/8 = 1/2

x = 8/2

x = 4

Answer 2

Using the angle bisector theorem the measure of side length x in the triangle is 4.

The figure in the image is a triangle separated by an angle bisector.

Note that, the angle bisector theorem states that "the angle bisector of a triangle separates or divides the opposite side into two segments that are proportional to the other two sides of the triangle".

From the figure:

AB = 12

AE = 8

BD = 6

DE = x

Using the angle bisector theorem:

AB/AE = BD/DE

Plug in the values:

12/8 = 6/x

Cross multiply:

12x = 8 × 6

12x = 48

x = 48/12

x = 4

Therefore, the value of x is 4.