Question

If the side of a rhombus is $20$ meters and its shorter diagonal is three fourth of its longer diagonal, then the area of the rhombus must be:

• A. $375m^2$
• B. $380m^2$
• C. $384m^2$
• D. $395m^2$

Answer 1

The correct option is C $384m^2$

In a rhombus, the diagonals bisect each other at ${90}^0$.

Referring the diagram, if $PR=a$ m, then $OP=\frac{a}{2}$ m

Given, $SQ=\frac{3}{4}PR=\frac{3a}{4}$ and $OQ=\frac{SQ}{2}=\frac{3a}{8}$

Since $△POQ$ is a right angled triangle,

${PQ}^2={OP}^2+{OQ}^2$

$\Rightarrow {20}^2={\left(\frac{a}{2}\right)}^2+{\left(\frac{3a}{8}\right)}^2$

$\Rightarrow 400=\frac{a^2}{4}+\frac{9a^2}{64}$

$\Rightarrow 400=\frac{25a^2}{64}$

$\Rightarrow a^2=1024$

$\Rightarrow a=32m$

Hence, length of the diagonals $=32m$ and $\frac{3\times 32}{4}=24m$

Area of a rhombus $=\frac{1}{2}\times (d_1\times d_2)=\frac{1}{2}\times 32\times 24=384m^2$