Problem: ABCD is a square with side of unit length. Points E and F are taken respectively on sides AB and AD so that AE=AF and the quadrilateral CDFE has maximum area. In square units this maximum area is:
Answer Choices:
A. 21​
B. 169​
C. 3219​
D. 85​
E. 32​
Solution:
Let x be the common length of AE and AF
Area (EGC)Area (DFEG)∴Area (CDFE)​=21​(1−x)(1)=x[2(1−x)+1​]=x(1−2x​)=21​(1+x−x2)=21​[45​−(x2−x+41​)]=85​−21​(x−21​)2​
The maximum value of this expression is 85​.
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