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 `12`. If side AC has a length of `27`, what is the length of side BC?

Answer 1

`bar(BC) = 33.6`

Explanation:

Applying sinus law to triangles `Delta ABD` and `Delta ADC` we have

`sin(alpha)/(bar(BD)) = sin(beta)/(bar(AB))`

and

`sin(alpha)/(bar(DC))=sin(gamma)/(bar(AC))`

but `gamma = pi - alpha` and `sin(gamma) = sin(alpha)` so the last equation reads now

`sin(alpha)/(bar(DC))=sin(beta)/(bar(AC))`

dividing the firts and last equation in both sides we have now

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

`bar(DC) = (bar(BD) bar(AC))/(bar(AB)) = (12 xx 27)/15 = 21.6`

Finally

`bar(BC) = bar(BD)+bar(DC) = 12+21.6 = 33.6`