Question

The quadrilateral formed by joining the mid-points of the sides of a quadrilateral $PQRS,$ taken in order$,$ is a rhombus$,$ if

• A. $PQRS$ is a rhombus
• B. $PQRS$ is a parallelogram
• C. diagonals of $PQRS$ are perpendicular
• D. diagonals of $PQRS$ are equal.

Answer 1

The correct option is D

diagonals of $PQRS$ are equal.

We know that $ABCD$ is a rhombus

So

$AB=BC=CD=DA$

Now,

Since $D$ and $C$ are midpoints of $PQ$ and $PS$

By midpoint theorem,

we get

$DC=\frac{1}{2}QS$

Since $B$ and $C$ are midpoints of $SR$ and $PS$

By midpoint theorem

we get

$BC=\frac{1}{2}PR$

Now, again, $ABCD$ is a rhombus

$\therefore BC=CD$

⇒ $\frac{1}{2}QS=\frac{1}{2}PR$

⇒ $QS=PR$

Hence, diagonals of $PQRS$ are equal

therefore, option $\left(D\right)$ is correct.