Question : The area of a rhombus having one side 10 cm and one diagonal 12 cm is:
Option 1: $48\text{ cm}^2$
Option 2: $96\text{ cm}^2$
Option 3: $144\text{ cm}^2$
Option 4: $192\text{ cm}^2$
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Correct Answer: $96\text{ cm}^2$
Solution :
The area of a rhombus, where $d_1$ and $d_2$ are the lengths of the diagonals,
$ = \frac{1}{2} \times d_1 \times d_2 $
In this case, one side of the rhombus is given as 10 cm and one diagonal is given as 12 cm. The other diagonal can be found using the Pythagorean theorem in the right triangle formed by a side and two halves of the diagonals.
Let the other diagonal as $d_2$.
$⇒ 4 \times \text{side}^2=d_1^2+d_2^2$
$⇒ 4 \times100=12^2+d_2^2$
$⇒ 4 00=144+d_2^2$
$⇒ d_2^2=4 00-144 $
$⇒d_2 = 16$
Substituting $d_1 = 12 \text{ cm}$ and $d_2 = 16 \text{ cm}$ into the formula,
The area of a rhombus,
$= \frac{1}{2} \times 12 \times 16 = 96 \text{ cm}^2 $
Hence, the correct answer is $96 \text{ cm}^2 $.
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