Question

ABCD is a rectangle and P, Q, R and Sare mid-points of the sides AB, BC, CD and DA
respectively. Show that the quadrilateral PQRS is a rhombus.​

Answer 1

Step-by-step explanation:

Given:ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively.

To Prove: PQRS is a rhombus.

Construction : Diagonals AC and BD are drawn.

Proof: In ∆ABC, P and Q are the mid-points of AD and BC. ∴ PQ || AC (Mid-point theorem) PQ = 1*2 1*2AC …………..

(i) Similarly, in ∆ADC, S and R are the mid-points of AD and CD. ∴ SR || AC SR = 1*21*2AC …………… (ii)

Similarly, in ∆ABD, SP || BD SP = 1*2 1*2BD ……………….. (iii) Similarly, in ∆BCD, QR || BD QR = 1*2 1*2BD ……………… (iv) From (i), (ii), (iii) and (iv), PQ = QR = SR = PS and Opposite sides are parallel.

∴ PQRS is a rhombus