Question

In the below Fig, 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

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

Comments

How is PQ parallel to AD?

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As we have given, PQRC is a rectangle which implies that PQ || RC  therefore, PQ || BC.
Hence, PQ || AD

Answer 2

Given, ABCD and PQRC are rectangles in which Q is mid-point of AC.

To Prove: DP=PC and PR=1/2AC

Proof:

Given, PQRC is a rectangle which implies that PQ || RC ⇒ PQ || BC.

In triangle BCD, we have

Q mid point of BD and PQ || BC.

Therefore, P is mid-point of CD. [Using converse of mid-point theorem which states that the line drawn through the midpoint

of one side of a triangle,parallel to another side, intersects the third side at its mid-point.]

⇒ DP = PC.

Again, in triangle BCD, P is mid-point of CD and PC || Qr which implies that DC || QR.

Again, R is mid-point of BC.  [By converse of mid-point theorem]

Again, in triangle BCD, P is mid-point of CD and R is mid-point of BC.

Therefore, PR = 1/2 BD. [Using mid-point theorem which states that the line segment joining the mid-points of two sides of a

triangle is parallel to the third side and half of it]

We know that both the diagonals of a rectangle are equal.

So, AC = BD.

⇒ PR = 1/2 AC

[Hence Proved]