Question

The diagonals of a quadrilateral ABCD are perpendicular, show that quadrilateral formed by joining the mid pts. of its sides, is a rectangle

Answer 1

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.