Question

Given a rectangle $ABCD$ and points $P,Q,R$ and $S$ are mid-points of sides $AB,BC,CD$ and $DA$ respectively. If length of a diagonal of the rectangle is $8cm,$ then the quadrilateral $PQRS$ is a _______.

• A. parallelogram with adjacent sides $4cm$ and $8cm$
• B. rhombus with side $4cm$
• C. rectangle with adjacent sides $4cm$ and $8cm$
• D. square with side $4cm$

Answer 1

The correct option is B

rhombus with side $4cm$

In $\Delta BCD$, the points $Q$ and $R$ are the mid-points of the sides $BC$ and $CD$ respectively.
$\therefore QR\parallel DB$

and $QR=\frac{1}{2}DB$ [By mid-point theorem]

Similarly, In $\Delta BAD$, the points $S$ and $P$ are the mid-points of the sides $AD$ and $BA$ respectively.
$\therefore SP\parallel DB$

and $SP=\frac{1}{2}DB$ [By mid-point theorem]

Hence can conclude from above that;

$QR\parallel PS=4cm$

Hence, $PQRS$ is a rhombus.