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Question

The midpoints of the sides AB, BC, CD and DA of a quadrilateral ABCD are joined to form a quadrilateral. If AC = BD and AC ⊥ BD then prove that the quadrilateral formed is a square.

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Solution



Given: In quadrilateral ABCD,
AC = BD and AC ⊥ BD. P, Q, R and S are the mid-points of AB, BC, CD and AD, respectively.

To prove: PQRS is a square.

Construction: Join AC and BD.

Proof:

In ΔABC,

P and Q are mid-points of AB and BC, respectively.


PQ || AC and PQ = 12AC (Mid-point theorem) ...(1)

Similarly, in ΔACD,

R and S are mid-points of sides CD and AD, respectively.

SR || AC and SR = 12AC (Mid-point theorem) ...(2)

From (1) and (2), we get

PQ || SR and PQ = SR

But this a pair of opposite sides of the quadrilateral PQRS.

So, PQRS is parallelogram.


Now, in ΔBCD,

Q and R are mid-points of sides BC and CD, respectively.

QR || BD and QR = 12BD (Mid-point theorem) ...(3)

From (2) and (3), we get

RS || AC and QR || BD


But, AC ⊥ BD (Given)

∴ RS ⊥ QR

But this a pair of adjacent sides of the parallelogram PQRS.

So, PQRS is a rectangle.


Again, AC = BD (Given)

12AC = 12BD

RS = QR [From (2) and (3)]

But this a pair of adjacent sides of the rectangle PQRS.

Hence, PQRS is a square.

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