Question

A rhombus has perimeter 64 m and one of the diagonals is 22 m, then the area of the rhombus is

• A. 255.66 $m^2$
• B. 255.56 $m^2$
• C. 255.44 $m^2$
• D. none of the above

Answer 1

The correct option is A 255.66 $m^2$

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 $=>4\times PQ=64m=>PQ=16m$

Hence, length of each side of the rhombus $=16m$
In a rhombus, the diagonals bisect each other at ${90}^0$

Referring the diagram, if PR $=22m$, then OP $=11m$, OQ $=\frac{SQ}{2}$

Since, triangle POQ is a right angled triangle,
${PQ}^2={OP}^2+{OQ}^2$
${16}^2={11}^2+{\left(\frac{SQ}{2}\right)}^2$

${\left(\frac{SQ}{2}\right)}^2={16}^2-{11}^2$
$\left(\frac{SQ}{2}\right)=\sqrt{135}=3\sqrt{15}m$
$SQ=6\sqrt{15}m$
Hence, length of the other diagonal $=6\sqrt{15}m$

Area of a rhombus $=\frac{1}{2}\times$ product of diagonals $=\frac{1}{2}\times 22\times 6\sqrt{15}=255.66m^2$