Question

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

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

Answer 1

The correct option is C

diagonals of $PQRS$ are perpendicular

Step $1:$Let the rectangle be $ABCD,$

$\angle DAB=90°$$.$

Consider $△PQR$

$A$ and $B$ are the mid points of sides $PQ$ and $QR$ respectively

$AB\parallel PR$ and $AB=\frac{1}{2}PR$ By mid point theorem$.$

Now we can say that $,$ $AT\parallel OP$

and $AT\parallel UO$

Similarly$,$

Step $2:$ In triangle $△PSQ$

$A$ and $D$ are the mid points of sides $PQ$ and $PS$ respectively

$AD\parallel QS$ and $AD=\frac{1}{2}SQ$ By mid point theorem$.$

Now we can say that $,$ $AU\parallel OQ$

and $UA\parallel OT$

Step $3:$In quadrilateral $AUOT$

$AT\parallel UO$ and $UA\parallel OT$ therefore quadrilateral $AUOT$ is a parallelogram

and angle $\angle UAT=\angle DAB=90°$

$\angle UAT=\angle UOT=90°$$,$ opposite angles of a parallelogram are equal

It shows that diagonal $PR$ and $QS$ are perpendicular to each other.

Hence, diagonals of $PQRS$ are perpendicular$,$ option $\left(C\right)$ is correct.