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

The figure is given below:

To Prove: PQRS is a rectangle.
For that, we need to prove that, PQ = RS, PS = QR and
also, that angles of PQRS are all right angles.

Proof:
We start by proving the equality of sides of quadrilateral PQRS

In ΔABC, P and Q are the mid-points of sides AB and BC respectively,
The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints
of those sides is parallel to the third side and is half the length of the third side.

∴ From the mid-point theorem,

PQ || AC &
PQ = ½ AC ..(1)

Also, R and S are the mid-points of CD and AD respectively

∴ From the mid-point theorem,

RS || AC
& RS = ½ AC ..(2)

Therefore,

From equations (1) and (2), we obtain

PQ || RS and PQ = RS

Since in quadrilateral PQRS, one pair of opposite sides is equal and parallel to each other
So it is a parallelogram.

Let the diagonals of rhombus ABCD intersect each other at point O

MQ || ON (Because PQ || AC)

QN|| OM (Because QR || BD)

Therefore,

OMQN is a parallelogram

MQN = NOM

PQR = NOM

However, NOM = 900 (Diagonals of a rhombus are perpendicular to each other)

So, PQR = 900

Clearly, PQRS is a parallelogram having one of its interior angles as 900

Hence, PQRS is a rectangle
Hence, Proved.

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