A triangle has corners at points A, B, and C. Side AB has a length of #21 #. The distance between the intersection of point A's angle bisector with side BC and point B is #7 #. If side AC has a length of #14 #, what is the length of side BC?

Let D be the point on BC where the angle bisector from A, intersects with BC

Then BC = BD + DC

We know BD = 7 and require to find DC.

substitute known values into the equation.

To find the length of side BC, you can use the Angle Bisector Theorem, which states that in a triangle, the length of the side opposite a given angle is proportional to the lengths of the other two sides.

Let (D) be the point where the angle bisector of angle (A) intersects side (BC). According to the given information, (AD) bisects angle (A).

Given that the length of side (AB) is 21 and the length of side (AC) is 14, we can set up the following proportion:

[\frac{BD}{DC} = \frac{AB}{AC}]

Substituting the given values, we get:

[\frac{BD}{DC} = \frac{21}{14} = \frac{3}{2}]

Now, let (x) be the length of (BD), so (DC) would be (2x) since (BD) and (DC) are in the ratio (3:2).

Since the distance between the intersection of point (A)'s angle bisector with side (BC) and point (B) is 7, we have:

[x + 7 = 21]

Solving for (x):

[x = 21 - 7 = 14]

Therefore, the length of side (BC), which is (BD + DC), is (14 + 2(14) = 14 + 28 = 42).