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Question

In given figure, ABCDE is any pentagon. BP is drawn parallel to AC and meets DC produced at P. EQ is drawn parallel to AD and meets CD produced at Q. Then, which of the following statements is true?


A

Area(ABCDE)=Area(ΔAPQ)

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B

Area(ABCDE)=Area(ΔACQ)

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C

Area(ABCDE)=2 Area(ΔADQ)

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D

None of these

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Solution

The correct option is A

Area(ABCDE)=Area(ΔAPQ)


BP || AC and AD || EQ,
Since triangles on the same base between the same parallels are equal in area
ar(ΔABC)=ar(ΔAPC) ........(1) (As Triangles that lie between two parallel lines and have same bases are equal in area.)
ar(ΔADE)=ar(ΔADQ) ..........(2) (As Triangles that lie between two parallel lines and have same bases are equal in area.)
adding (1) and (2) we get
ar(ΔABC)+ar(ΔADE)=ar(ΔAPC)+ar(ΔADQ)
Adding ar(ΔACD) to both sides, we get
ar(ΔABC)+ar(ΔADE)+ar(ACD)=ar(ΔAPC)+ar(ΔADQ)+ar(ACD)
Hence ar(ABCDE)=ar(ΔAPQ)


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