Question : The height of an equilateral triangle is 18 cm. Its area is:
Option 1: $36\sqrt{3}\operatorname{cm^2}$
Option 2: $108\sqrt{3}\operatorname{cm^2}$
Option 3: $108\operatorname{cm^2}$
Option 4: $96\sqrt{3}\operatorname{ cm^2}$
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Correct Answer: $108\sqrt{3}\operatorname{cm^2}$
Solution :
Given that the height ($h$) of the equilateral triangle is 18 cm.
$ h = \frac{\sqrt{3}}{2} \times a $
$ a = \frac{2h}{\sqrt{3}} = \frac{2 \times 18}{\sqrt{3}} = 12\sqrt{3}\operatorname{ cm}$
The area (A) of an equilateral triangle,
$ A = \frac{\sqrt{3}}{4} \times a^2 $
On substituting $a = 12\sqrt{3}$ cm,
$ A = \frac{\sqrt{3}}{4} \times (12\sqrt{3})^2 = \frac{\sqrt{3}}{4} \times 432 = 108\sqrt{3}\operatorname{ cm^2}$
Hence, the correct answer is $108\sqrt{3}\operatorname{ cm^2}$.
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