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Question

ABCD is a parallelogram. XandY are mid-points of BCandCD respectively. Then area (ΔAXY)=3k area(parallelogram ABCD).Find k

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Solution

Given:ABCDisaparallelogram.XandYaremidpointsofBCandCDrespectively.SinceXandYarethemidpointsofsidesBCandCDrespectivelyinBCDXYBDandXY=12BD.Area(CYX)=14area(DBC)=18area(parallelogramABCD)....(1)[Sincear(DBC)=12area(parallelogram(ABCD)]ParallelogramABCDandABXarebetweenthesameparallelsADandBCandBX=12BCarea(ABX)=14area(parallelogramABCD).....(2)Similarly,area(AYD)=14area(parallelogramABCD)Now,ar(AXY)=ar(parallelogramABCD)[ar(ABX)+ar(AYD)+ar(CYX)]ar(AXY)=ar(parallelogramABCD)(14+14+18)ar(ABCD)ar(AXY)=(158)ar(parallelogramABCD)=38ar(parallelogramABCD).
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