Question

The perimeter of a rhombus is $146cm$. One of its diagonals is $55cm$. Then the length of the other diagonal and the area of the rhombus is

• A. $48cm,1320c{m}^2$
• B. $45cm,660c{m}^2$
• C. $27.5cm,660c{m}^2$
• D. None of these

Answer 1

Answer: (A)

$48cm,1320c{m}^2$

Let $PQRS$ be the rhombus as shown in the figure.

Since the length of all sides of a rhombus are equal, Perimeter $=4\times$ length of one side $\Rightarrow 4\times PQ=146cm\Rightarrow PQ=36.5cm$

Hence, length of each side of the rhombus $=36.5cm$
In a rhombus, the diagonals bisect each other at ${90}^0$

Referring the diagram, if $PR=55cm$, then $OP=27.5cm$, $OQ=\frac{SQ}{2}$

Since, triangle $POQ$ is a right angled triangle,

${PQ}^2={OP}^2+{OQ}^2$
${36.5}^2={27.5}^2+{\left(\frac{SQ}{2}\right)}^2$

${\left(\frac{SQ}{2}\right)}^2={36.5}^2-{27.5}^2$
$\left(\frac{SQ}{2}\right)=\sqrt{576}=24cm$
$SQ=48cm$
Hence, length of the other diagonal $=48cm$

Area of a rhombus $=\frac{1}{2}\times (\text{diagonal 1}\times \text{diagonal 2})=\frac{1}{2}\times 55\times 48=1320{cm}^2$