Problem:
In triangle ABC, side AC and the perpendicular bisector of BC meet in point D, and BD bisects β ABC. If AD=9 and DC=7, what is the area of triangle ABD ?
Answer Choices:
A. 14
B. 21
C. 28
D. 145β
E. 285β Solution:
By the angle-bisector theorem, BCABβ=79β. Let AB=9x and BC=7x, let mβ ABD=mβ CBD=ΞΈ, and let M be the midpoint of BC. Since M is on the perpendicular bisector of BC, we have BD=DC=7. Then
cosΞΈ=727xββ=2xβ
Applying the Law of Cosines to β³ABD yields
92=(9x)2+72β2(9x)(7)(2xβ)
from which x=4/3 and AB=12. Apply Heron's formula to obtain the area of triangle ABD as 14β 2β 5β 7β=145ββ.
