ABCD is a parallelogram X and Y are midpoints of BC and CD respectively. Prove that ar(triangle AXY)= 3/8 ar(parallelogram ABCD)

Asked by Sthitaprajna Mishra | 4th Feb, 2014, 08:11: PM

Expert Answer:

Consider the following figure.
 
 
Join the diagonals AC and BD to bisect at O.
 
OY produced meets AB at M and OX produced meets AB at N.
 
Since O and Y are the midpoints of AC and CD, we have fraction numerator O Y over denominator A D end fraction equals 1 half
Thus, we have OY = CX
 
Therefore OXCY is a parallelogram.
 
Since parallelograms with the same base and between the same parallels are of equal areas, we have
 
a r open parentheses square O X C Y close parentheses equals a r open parentheses square O X B M close parentheses equals a r open parentheses square O M A N close parentheses equals a r open parentheses square O N D Y close parentheses
Also, we have a r open parentheses square O X C Y close parentheses equals 1 fourth a r open parentheses square A B C D close parentheses
Now consider the traingle CXY.
 
Thus,
a r open parentheses triangle C X Y close parentheses equals 1 half a r open parentheses square O X C Y close parentheses rightwards double arrow a r open parentheses triangle C X Y close parentheses equals 1 half open parentheses 1 fourth a r open parentheses square A B C D close parentheses close parentheses rightwards double arrow a r open parentheses triangle C X Y close parentheses equals 1 over 8 begin inline style a end style begin inline style r end style begin inline style open parentheses square A B C D close parentheses end style
Now a r open parentheses triangle A C D close parentheses equals 1 half open parentheses a r square A B C D close parentheses
Since Y is the midpoint of CD, we have, a r open parentheses triangle A Y D close parentheses equals 1 half a r open parentheses triangle A C D close parentheses equals 1 fourth a r open parentheses square A B C D close parentheses
In the same way, a r open parentheses triangle A B X close parentheses equals 1 fourth a r open parentheses square A B C D close parentheses
 
Now consider the area of the parallelogram ABCD.
 
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Answered by  | 6th Feb, 2014, 12:10: PM

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