Question

In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ. Show that AQ = CP.

Answer 1

Hint: Draw the diagonal AC and assume that it intersects the diagonal BD at O. To prove that AQ = CP, prove that quadrilateral AQCP is also a parallelogram by proving PO = QO. Use the property of parallelogram that ‘diagonals of a parallelogram bisect each other’.

Complete step-by-step solution -

Let us construct or join the diagonal AC of the parallelogram which intersects the diagonal BD at O. we know that diagonals of a parallelogram bisect each other. Therefore, BO = DO and AO = CO.

We have been given that, DP = BQ. Therefore,
$\begin{align}
  & BO-BQ=DO-DP \\
 & \Rightarrow OQ=OP \\
\end{align}$
Hence, we can say that the diagonal QP of the quadrilateral AQCP is bisected at point O. Also, it is already known that AO = CO. Therefore, the conclusion is that diagonals AC and PQ of the quadrilateral AQCP bisect each other. Hence, AQCP is a parallelogram.
Now, we know that opposite sides of a parallelogram are equal. Therefore, AQ = CP.

Note: One may note that there are certain quadrilaterals whose diagonals bisect each other but here we have assumed the quadrilateral as a parallelogram because this is the basic property of a parallelogram. Other special quadrilaterals have certain other properties which we are not able to prove here because of limited information. So, if we have to prove that a quadrilateral is a parallelogram, we just prove that its diagonals bisect each other.