Question

A quadrilateral, $ABCD$ is drawn in which the mid points of sides $AB,BC,CD$ and $AD$ are $P,Q,R$ and $S$ respectively.

If quadrilateral, $ABCD$ is a parallelogram, what can you say about the quadrilateral, $PQRS?$

• A. Parallelogram
• B. Rhombus
• C. Rectangle
• D. Can’t say

Answer 1

The correct option is A

Parallelogram

Draw the diagonals of the parallelogram as shown in the figure below:

From mid-point theorem:

In $△ABD,$

$SP\parallel BD$ and

$SP=\frac{1}{2}BD$ ... (i)

In $△BCD,$

$QR\parallel BD$ and

$QR=\frac{1}{2}BD$ ... (ii)

In $△ABC,$

$PQ\parallel AC$ and

$PQ=\frac{1}{2}AC$ ... (iii)

In $△ADC,$

$RS\parallel AC$ and

$RS=\frac{1}{2}AC$ ... (iv)

Therefore, $SP=QR$ [from (i) and (ii)]

and $PQ=RS$ [from (iii) and (iv)]

Since, the opposite sides are equal and parallel. So, $PQRS$ is a parallelogram.

We need to check whether one of the angles is ${90}^{\circ }$ or not to determine if $PQRS$ is a rectangle or not.

Since, $RS\parallel AC$ and $QR\parallel BD,\angle AOB=\angle SRQ$

We know that diagonals, $AC$ and $BD$ bisect each other and do not intersect at right angles.

Therefore, $PQRS$ is a parallelogram and not a rectangle.