Problem:
Points E and F are located on square ABCD so that △BEF is equilateral. What is the ratio of the area of △DEF to that of △ABE?
Answer Choices:
A. 34
B. 23
C. 3
D. 2
E. 1+3
Solution:
First, assume that AB=1, and let ED=DF=x. By the Pythagorean Theorem x2+x2=EF2=EB2=12+(1−x)2, so x2=2(1−x). Hence the desired ratio of the areas is
Area(△ABE)Area(△DEF)=1−xx2=2
Answer: D.
The problems on this page are the property of the MAA's American Mathematics Competitions
