Question

ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.
step by step explaination
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Answer 1

In △ABC, P and Q are the mid-points of sides AB and BC.

Using Mid-point Theorem,

PQ∥AC and PQ= 1/2AC ---(1)

Similarly,

In △ADC, R and S are the mid-points of sides CD and AD.

Using Mid-point Theorem

SR∥AC and SR= 1/2AC ---(2)

From (1) and (2), we get

PQ∥SR and PQ=SR=1/2AC ---(3)

similarly, PS∥QR and PS=QR=1/2BD ---(4)

Also,

AC=BD [diagonals of a rectangle are equal]

⇒1/2AC= 1/2BD

⇒PQ=SR=PS=QR ....from(3) and (4)

∴PQRS is a Rhombus