Question

In a rectangle ABCD, points X and Y are the midpoints of AD and DC, respectively. Lines BX and CD when extended intersect at E, lines BY and AD when extended intersect at F. If the area of ABCD is 60 then the area of BEF is

• A. 60
• B. 80
• C. 90
• D. 120

Answer 1

The correct option is C 90

$AB=x,BC=yxy=60XA=XD,\angle XAB=\angle XDE,\angle EXD=\angle AXB\therefore △XDE\cong △XAB\therefore ar△XDE=ar△XAB$

Similarly

$△YDF\cong △YCB\therefore ar△YDF=ar△YCB$

Also, $\angle FDY=\angle FAB,\angle DFY=DFY$

$\therefore △AFB\sim △DFY\therefore \frac{AF}{DF}=\frac{FB}{FY}=\frac{AB}{DY}\frac{AF}{DF}=\frac{FB}{FY}=\frac{x}{\frac{x}{2}}\Rightarrow AF=2DFAD=DF=y$

Similarly $DE=y$

Now, $ar△BEF=ar△YDF+ar△XDE+ar\square DXYB+ar△DEF$

$=ar△YCB+ar△XAB+ar\square DXYB+ar△DEF=ar\square ABCD+ar△DEF=xy+\frac{1}{2}\times DF\times DE=xy+\frac{xy}{2}=60+\frac{60}{2}=90$