Problem:
In triangle ABC,∠CBA=72∘,E is the middpoint of side AC, and D is a point on side BC such that 2BD=DC;AD and BE intersect at F. The ratio of the area of △BDF to the area of quadrilateral FDCE is
Answer Choices:
A. 51​
B. 41​
C. 31​
D. 52​
E. none of these
Solution:
In the adjoining figure the line segment from E to G, the midpoint of DC, is drawn. Then
​ area △EBG=(32​)( area △EBC) area △BDF=(41​)( area △EBG)=(61​)( area △EBC)​
(Note that since EG connects the midpoints of sides AC and DC in △ACD, EG is parallel to AD). Therefore, area FDCE=(65​)( area △EBC) and
( area FDCE)( area △BDF)​=51​
The measure of ∠CBA was not needed.
