Question

A triangle has corners at points A, B, and C. Side AB has a length of `42`. The distance between the intersection of point A's angle bisector with side BC and point B is `16`. If side AC has a length of `36`, what is the length of side BC?

Answer 1

`a = { 16 (36+42)} / 42 = 208/7`

Explanation:

Thanks for using upper case for the vertices.

Let me gently urge you folks out there toward standard terminology and notation. The corners of a triangle are its vertices, each is a vertex. The usual convention is small letters for the sides, so `AB=42` may be written `c=42`. We call the intersection of various bisectors, medians or perpendiculars with their opposite side the "foot" of the .e.g. angle bisector.

So we can write this question:

Triangle ABC has `c=42, b=36.` Call F on BC the foot of the angle bisector of A. Given `d=BF=16`, find `a`.

Call `e=CF`. By the angle bisector theorem

`{AB}/{ BF} = {AC}/{CF} quad or quad c/d=b/e`

We really after `a=BC=BF+CF=d+e` so let's substitute `e=a-d.`

`c (a-d) = bd`

`ac = bd + cd`

`a = {d(b+c)}/c`

`a = { 16 (36+42)} / 42 = 208/7`

Check:

`{AB}/{BF} = c/d=42/16=21/8`

`{AC}/{CF} = b/e = 36/(208/7 - 16) = 21/8 quad sqrt`