Question

ABCD is a trapezium in which AB$||$DC and AD$=$BC. If P, Q, R, S be the mid-points of BA, BD, CD and CA respectively. What is the shape of PQRS?

• A. Rhombus
• B. Rectangle
• C. Parallelogram
• D. Square

Answer 1

The correct option is A Rhombus

In $△ABC$, $P,S$ are the mid points of $AB,AC$ respectively.

So, $PS||BC$ and $PS=\frac{1}{2}BC$ ---(i)

Similarly, in $△ADC$,

$RS=\frac{1}{2}AD=\frac{1}{2}BC$ ---(ii)

In $△BCD$,

$RQ=\frac{1}{2}BC$ ---(ii)

Similarly, in $△ABD$,

$QP=\frac{1}{2}AD=\frac{1}{2}BC$ ---(iv)

$\therefore From\left(i\right)\left(ii\right)\left(iii\right)\left(iv\right)$, we get

$PQ=QR=RS=SP$

Since all sides are equal, $PQRS$ is a rhombus

Option A.