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Question

In the given figure ABCD is a parallelogram. If P is a point on the diagonal, which of the following is true?


  1. ar(APD)=ar(APB)
  2. ar(APD)=ar(PDC)
  3. ar(APD)=ar(BPC)
  4. ar(CPD)=ar(APB)


Solution

The correct option is A ar(APD)=ar(APB)
Let’s draw perpendicular from B and D to the diagonal.
Take ADQ and CBR
Q=R  (each 90)
A=C  ( alternate interior angles)
So, D=B (third angle of triangle)
AD = BC ( opposite sides of parallelogram)
Using ASA criteria
ADQCBR
So, DQ=BR
ar(APD)=12×AP×DQ
ar(APB)=12×AP×BR
We know that DQ = BR
Hence ar(APD)=ar(APB)

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