Problem:
Convex quadrilateral ABCD has AB=9 and CD=12. Diagonals AC and BD intersect at E,AC=14, and â–³AED and â–³BEC 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=(E)6​.
The problems on this page are the property of the MAA's American Mathematics Competitions
