Question

The length of the side of a rhombus is 10 units and its diagonals differ by 4. The area of the rhombus is

• A. $108$
• B. $96$
• C. $84$
• D. $48$

Answer 1

The correct option is B $96$

Let $d_1$ and $d_2$ be the lengths of the diagonals of the rhombus.
Then, $d_1+d_2=4$
In a rhombus, the diagonals bisect each other at ${90}^0$
Referring the diagram, if PR $=d_1$, SQ $=d_2$, then OP $=\frac{d_1}{2}$, OQ $=\frac{d_2}{2}$
Since, triangle POQ is a right angled triangle,
${PQ}^2={OP}^2+{OQ}^2$
${10}^2={\left(\frac{d_1}{2}\right)}^2+{\left(\frac{d_2}{2}\right)}^2$
$100=\frac{d_1{}^2}{4}+\frac{d_2{}^2}{4}$
$400=d_1{}^2+d_2{}^2$
$400={d_1+d_1}^2-2d_1{d}_2$
$400=4^2-2d_1{d}_2$
$2d_1{d}_2=384$
$d_1{d}_2=192$

Area of a rhombus $= \dfrac { 1 }{2 } \times (diagonal 1 \times diagonal 2) =\dfrac{1}{2}{ d }_{ 1 }{ d }_{ 2 }=\dfrac { 1 }{ 2 } \times 192 = 96$