Question

ABCD is a rectangle and P Q R S are the midpoints of sides a
AB BC CD and DArespectively prove that pqrs is a rhombus

Answer 1

Let us join AC and BD.

In triangle ABC,
P and Q, are the mid point of AB and BC respectively.
since, PQ || AC and PQ = AC ( mid point theorem ) -------------(1)

similarly,
In triangle ADC,
SR || AC and SR = AC -------------(2)

From eq (1) and (2),
PQ || SR and PQ= SR

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

PS || QR and PS = QR ( opposite sides of parallelogram ) ----------- (3).

In triangle BCD, Q and R mid points of sides BC and CD respectively
since, QR || BD and QR = BD ( Mid point theorem)---------- (4)

However the diagnols of rectangle are equal.

AC = BD -----------(5).

By using eq (1), (2) , (3), (4), (5).
PQ = QR = RS = SP

therefore, it is rhombus

HENCE PROVED.....

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