Question

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

Answer 1

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To Prove:

PQRS is a rhombus.

Given:

ABCD is a rectangle and P, Q, R, and S are midpoints of AB, BC, CD, and DA.

Explanation:

PQ || AC and PQ = 1/2 AC (Mid-point theorem) (1)

Similarly, in ΔADC,

SR || AC and SR = 1/2 AC (Mid-point theorem) (2)

Clearly,

PQ || SR and PQ = SR

PS || QR and PS = QR (Opposite sides of a parallelogram) (3)

In ΔBCD, Q and R are the mid-points of side BC and CD respectively.

QR || BD and QR = 1/2 BD (Mid-point theorem) (4)

However, the diagonals of a rectangle are equal.

AC = BD (5)

By using equation (1), (2), (3), (4), and (5), we obtain

PQ = QR = SR = PS

Now, as all sides of the rhombus are equal.

Hence, PQRS is a rhombus.

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