Problem:
Convex quadrilateral ABCD has AB=9 and CD=12. Diagonals AC and BD intersect at E,AC=14, and △AED and △BEC have equal areas. What is AE?
Answer Choices:
A. 29
B. 1150
C. 421
D. 317
E. 6
Solution:
Because △AED and △BEC have equal areas, so do △ACD and △BCD. Side CD is common to △ACD and △BCD, so the altitudes from A and B to CD have the same length. Thus AB∥CD, so △ABE is similar to △CDE with similarity ratio
ECAE=CDAB=129=43
Let AE=3x and EC=4x. Then 7x=AE+EC=AC=14, so x=2, and AE=3x=6.
The problems on this page are the property of the MAA's American Mathematics Competitions
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