Question

ABCD is a rhombus and P, Q, R, S are mix points of AB, BC, CD and AD respecticely. show that quadrilateral PQRS is a rectangle.
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Answer 1

Given :

P, Q , R and S are the midpoint of respectively side AB , BC , CD and DA of rhombus PQ , QS , SR , PS are joined

To Prove:

PQRS is a rectangle.

Construction:

Join A and C.

Proof:

In ΔABC, P is the mid-point of AB and Q is the mid-point of BC.

By Mid-point theorem,

PO || AC and PQ = ½AC ..... (1)

In ΔADC, R is the mid-point of CD and S is the mid-point of AD.

By Mid-Point theorem,

SR || AC and SR = ½AC ... (2)

From (1) and (2), we get

PQ || SR and PQ = SR

PQRS a parallelogram.

[In quad. PQRS one pair of opposite sides PQ and SR are parallel and equal]

Now ABCD is a rhombus

AB = BC

½ AB = ½ BC

PB = BQ

angle 1 = angle 2

[angle s opposite to equal sides of a triangle are equal]

Now in Δs APS and COR,

AP = CQ

[: AB = BC = ½AB = ½BC = AP = CQ where P and Q are mid-points of AB and BC]