Question

P,Q,R,R, and S are respectively the mid-points of sides AB, BC, CD, And DA of a quadrilateral ABCD in which AC =BD, then PQRS is a rhombus.

• A. True
• B. False

Answer 1

The correct option is A True

$P,Q,R$ and $S$ are the mid-point of the sides $AB,BC,CD$ and $DA$ of a quadrilateral $ABCD$.

$\Rightarrow$ $AC=BD$

In $△ABC,$

$P$ and $Q$ are the mid-points of the sides $AB$ and $BC$ respectively.

$\therefore$ $PQ\parallel AC$ ----- ( 1 )

And $PQ=\frac{1}{2}\times AC$ ------ ( 2 )

Similarly, $SR\parallel AC$ and $SR=\frac{1}{2}\times AC$ ----- ( 3 )

From ( 1 ), ( 2 ) and ( 3 ) we get,

$\Rightarrow$ $PQ\parallel SR$ and $PQ=SR=\frac{1}{2}\times AC$ ----- ( 4 )

Similarly we an show that,

$\Rightarrow$ $SP\parallel RQ$ and $SP=RQ=\frac{1}{2}\times BD$ ----- ( 5 )

Since, $AC=BD$

$\therefore$ $PQ=SR=SP=RQ$ [ From ( 4 ) and ( 5 ) ]

All sides of the quadrilateral are equal.

$\therefore$ $PQRS$ is a rhombus.