Question

If $P(2,-1),Q(3,4),R(-2,3)$ and $S(-3,-2)$ are four points in a plane, show that $PQRS$ is a rhombus but not a square. Also, find its area.

• A. $24$ sq units
• B. $16$ sq units
• C. $48$ sq units
• D. $44$ sq units

Answer 1

The correct option is A $24$ sq units

$PQ=\sqrt{(3-2{)}^2+(4+1{)}^2}=\sqrt{1+25}=\sqrt{26}$ units
$QR=\sqrt{(-2-3{)}^2+(3-4{)}^2}=\sqrt{25+1}=\sqrt{26}$ units
$RS=\sqrt{(-3+2{)}^2+(-2-3{)}^2}=\sqrt{1+25}=\sqrt{26}$ units
$PS=\sqrt{(-3-2{)}^2+(-2+1{)}^2}=\sqrt{25+1}=\sqrt{26}$ units
$PR=\sqrt{(-2-2{)}^2+(3+1{)}^2}=\sqrt{16+16}=\sqrt{32}=4\sqrt{2}$ units
$QS=\sqrt{(-3-3{)}^2+(-2-4{)}^2}=\sqrt{36+36}=\sqrt{72}=6\sqrt{2}$ units
Now we can see that,
$PQ=QR=RS=PS=\sqrt{26}$ units but $PR\ne QS$
$\Rightarrow PQRS$ is a quadrilateral whose all sides are equal but diagonals are not equal.
$\Rightarrow PQRS$ is a rhombus, not a square.
$\therefore$ Area of rhombus PQRS $=\frac{1}{2}\times$ (Product of diagonals)
$\Rightarrow \frac{1}{2}\times PR\times QS$ $=\frac{1}{2}\times 4\sqrt{2}\times 6\sqrt{2}=24$ sq.units