Question

In Fig. 14.100, ABCD and PQRC are rectangles and Q is the mid-point of AC. Prove that
(i) DP = PC (ii) PR = 1/2 AC.

Answer 1

Given

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

To Prove

i) DP = PC ii) PR = 1/2 AC

Construction

Join BD and PR

Proof

PQRC is a rectangle. So,

PQ || CR and QR || PC.

Q is a mid point in triangle BCD such that PQ is a perpendicular on line CD.

→ PQ || BC

→ Also, Q is a mid point on line BD.

So, by converse of BPT.

P is mid-point on line DC.

Therefore, DP = PC

ii) Now, P is mid-point on line DC. So, PC || QR

Also, DC || QR

Also, R is mid- point on line BC. So, PQ || CR

Again, In ∆BCD

P is mid-point on line CD and R is a mid-point on line BC.

→ PR = 1/2 BD

Also, the diagonals of the rectangle are equal

Therefore, PR = 1/2 AC

Hence, proved.