Question

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

Answer 1

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.