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Q.

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

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a

PQRS is rhombus

b

PQRS is rectangle

c

PQRS is square

d

PQRS is cube

answer is A.

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Detailed Solution

Here we have to join A and C
In Δ ABC
⇒ P is the mid point of AB
⇒ Q is the mid point of BC
then PQ || AC and PQ =

\frac{AC}{2}

In Δ ADC,
⇒ R is the midpoint of CD
S is the midpoint of AD
RS || AC,

So, PQ || RS = RS = PQ (one pair of opposite sides are equal)
In Δ APS and Δ BPQ
AP = BP [P is the midpoint of AB]
∠ PAS = ∠ PBQ all (90°)
AS = BQ
∴ Δ APS ≅ Δ BPS (SAS Congruence)
∴ PS = PQ
∴ BS = PQ and PQ = RS (opposite sides of parallelogram)
∴ PQ = RS = PS = RQ
∴ PQRS is a parallelogram with all sides are equal
∴ PQRS is a rhombus.

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