Question

In the figure, ABCD and PQRC are rectangles and Q is the mid-point of AC.

Prove that (i) DP = PC (ii) $PR=\frac{1}{2}AC$.

Answer 1

Given : ABCD are PQRC are rectangles and Q is the mid-point of AC.

To prove : (i) DP = PC (ii) $PR=\frac{1}{2}AC$

Construction : Join diagonals AC which passes through Q and join PR.

Proof : (i) In $\Delta ACD,$

Q is mid -point of AC and QP || AD (Sides of rectangle)

$\therefore$ P is mid-point of CD

$\therefore$ DP = PC

(ii) $\because$ PR and QC are the diagonals of rectangle PQRC

$\therefore$ PR = QC

But Q is the mid-point of AC

$\therefore QC=\frac{1}{2}AC$

Hence $PR=\frac{1}{2}AC$