Question

ABCD is a rectangle and P,Q,R and S are mid point of the sides AB ,BC,CD and DA respectively show that the quadrilateral PQRS is a rhombus ?
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Answer 1

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Explanation:

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Answer 2

$\LARGE{\underline{\frak{\purple{Answєr :}}}}$

Data : ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively.

To prove : PQRS is a rectangle.

Construction : Diagonals AC and BD are drawn.

Proof : To prove PQRS is a rectnagle, one of its angle should be right angle.

In ∆ADC, S and R are the mid points of AD and DC.

∴ SR || AC

SR = 1212AC (mid-point formula)

In ∆ABC, P and Q are the mid points AB and BC.

∴ PQ || AC PQ = ½AC.

g ∴ SR || PQ and SR = PQ

$\small{\boxed{\sf{{∴PQRSisaparallelogram.}}}}$

But diagonals of a rhombus bisect at right angles. 90° angle is formed at ‘O’.

∴ ∠P = 90°

∴ PQRS is a parallelogram, each of its angle is right angle.

This is the property of rectangle.

$\small{\boxed{\sf{{∴ PQRS is a rectangle}}}}$

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