Question

The diagonals of a quadrilateral ABCD are perpendicular. The quadrilateral formed by joining the mid points of its sides, is ?

• A. Trapezium
• B. Rectangle
• C. Square
• D. Rhombus

Answer 1

Answer: (B)

Rectangle

From above figure,

AC, BD are diagonals of quadrilateral ABCD are perpendicular.

Points P, Q, R and S are mid-points of AB and BC respectively.

In $△$ABC,

P and Q are mid points of AB and BC.

So, by Mid-point theorem,

$\therefore$ PQ$\left|\right|$AC; PQ = $\frac{1}{2}$AC -------- (1)

Similarly in $△$ACD,

R and S are mid point of sides CD and AD.

$\therefore$ SR$\left|\right|$AC; SR = $\frac{1}{2}$AC ----------- (2)

From (1) and (2), we get

PQ$\left|\right|$SR and PQ = SR

Hence, PQRS is parallelogram -------( pair of opposite sides are equal and parallel)

Now,RS$\left|\right|$AC and QR$\left|\right|$BD.

Also, AC$\perp$BD ---- (Given)

$\therefore$ RS$\perp$QR.

Thus, PQRS is a rectangle.