Question

ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see the given figure). AC is a diagonal. Show that: PQRS is a rectangle.

Answer 1

Join AC and AD such that it intersect it at O.

In triangle ABD
by mid point theorem
SP || BD. ( means parallel )
SP = HALF BD ( 1 )

by mid point theorem
RQ || BD
RQ = HALF BD ( 2 )

FROM ( 1 ) ( 2 )
SP = RQ

PQRS IS PARALLELOGRAM
SM || TO & ST || MO
similarly
SMOT IS PARALLELOGRAM
one angle is 90°
thus PQRS is ractangle

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