Question

If the sum of the lengths of the diagonals of a rhombus of side $4cm$ is $10cm$. What is its area?

• A. $8c{m}^2$
• B. $9c{m}^2$
• C. $10c{m}^2$
• D. $12c{m}^2$

Answer 1

The correct option is B $9c{m}^2$

Let $d_1$ and $d_2$ be the lengths of the diagonals of the rhombus. Then,
${\left(\frac{d_1}{2}\right)}^2+{\left(\frac{d_2}{2}\right)}^2=4^2\Rightarrow \frac{d_2^2}{4}=16$
$\Rightarrow d_1^2+d_2^2=64$
$\Rightarrow (d_1+d_2{)}^2-2d_1{d}_2=64$
$\Rightarrow {10}^2-2d_1=100-64$ $(\therefore d_1+d_2=10)$
$\Rightarrow 2d_1{d}_2=100-64=36$
$\Rightarrow d_1{d}_2=18\Rightarrow \frac{1}{2}{d}_2{d}_2=9c{m}^2$
$\therefore$ Area of the rhombus $=9c{m}^2$