Question

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

Answer 1

$\huge\fbox \pink{A}\fbox \purple{n}\fbox \blue{s}\fbox \pink{w}\fbox \purple{e}\fbox \blue{r}$

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

$\small{\boxed{\sf{{∴ PQRS is a parallelogram. }}}}$

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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