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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