Question

A triangle has corners at points A, B, and C. Side AB has a length of `15`. The distance between the intersection of point A's angle bisector with side BC and point B is `8`. If side AC has a length of `15`, what is the length of side BC?

Answer 1

Length of side `BC = 16`

Explanation:

Let the point where the angle bisector intersects with
side BC be D

`"using the "color(blue)"angle bisector theorem"`

`(AB)/(AC)=(BD)/(DC)`

`15 / 15 = 8 / (DC`

`DC = (8*cancel(15)) / cancel(15) = 8`

It’s an isosceles triangle with sides AB & AC equal.

Hence BD = DC & BC = BD + DC = 2*BD

`BC = BD+DC= 8+8 =16`

Answer 2

`BC=16`

Explanation:

`"let the point where the angle bisector intersects BC be D"`

`"using the "color(blue)"angle-bisector theorem"" then"`

`color(red)(bar(ul(|color(white)(2/2)color(black)((AB)/(AC)=(BD)/(DC))color(white)(2/2)|)))`

`rArr15/15=8/(DC)larrcolor(blue)"cross-multiply"`

`15xxDC=8xx15`

`rArrDC=(8xx15)/15=8`

`rArrBC=BD+DC=8+8=16`