8. In the given figure, ABCD is a quadrilateral whose diagonals are
perpendicular to each other. P, Q, R and S are mid-points of sides AB,
BC, CD and DA respectively. Show that quadrilateral PQRS is a rectangle.
Answers
Answer:
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Step-by-step explanation:
Given: A quadrilateral ABCD whose diagonals AC and BD are perpendicular to each other at O. P,Q,R and S are mid points of side AB, BC, CD and DA respectively are joined are formed quadrilateral PQRS.
To Prove : PQRS is a rectangle.
Proof : In △ABC, P and Q are mid - points of AB and BC respectively.
∴ PQ || AC and PQ = 1 / 2 AC ---- (i) [mid point theorem]
Further, in △ACD, R and S are mid points of CD and DA respectively.
∴ SR || AC and SR = 1 / 2 AC --- (ii) [mid point theorem]
From (i) and (ii) , we have PQ || SR and PQ = SR
Thus , one pair of opposite sides of quadrilateral PQRS are parallel and equal .
∴ PQRS is a parallelogram .
Since PQ || AC ⇒ PM || NO
In △ABD, P and S are mid points of AB and AD respectively .
∴ PS || BD [mid point theorem]
⇒ PN || MO
∴ Opposite sides of quadrilateral PMON parallel .
∴ PMON is a parallelogram .
∴ ∠MPN = ∠MON [opposite angles of || gm are equal]
But ∠MON = 90° [give]
∴ ∠MPN = 90° ⇒ ∠QPS = 90°
Thus, PQRS is a parallelogram whose one angle is 90°.
∴ PQRS is a rectangle.