In the adjoining figure, ABCD and PQRC are rectangles, where Q is the midpoint of AC. Prove that
(i) `DP=PC, " (ii) " PR=(1)/(2) AC. `

Explanation

`angle CRQ = angle CBA =90^(@) rArr `QR||AB.
In `triangle ABC`, Q is the midpoint of AC and QR||AB.
`therefore ` R is the midpoint of BC, i.e., BR = RC.
Similarly, P is the midpoint of DC.
`therefore DP =PC.`
In `triangle CDB`, P is the midpoint of DC and R is the midpoint of BC.
`therefore " PR||DB and " PR=(1)/(2)DB=(1)/(2)AC " " [ because AC = BD].`