Question : If the height of the equilateral triangle is $2 \sqrt 3\:\operatorname{cm}$, then determine the area of the equilateral triangle.
Option 1: $6\:\operatorname{cm^2}$
Option 2: $2\sqrt3\:\operatorname{cm^2}$
Option 3: $4\sqrt3\:\operatorname{cm^2}$
Option 4: $12\:\operatorname{cm^2}$
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Correct Answer: $4\sqrt3\:\operatorname{cm^2}$
Solution :
In an equilateral triangle, the height $(h)$ is related to the side length $(a)$,
$h = \frac{\sqrt{3}}{2}a$
Given that the height of the equilateral triangle is $2\sqrt{3}\:\text{cm}$.
⇒ $2\sqrt{3} = \frac{\sqrt{3}}{2}a$
⇒ $a = \frac{2\sqrt{3} \times 2}{\sqrt{3}} = 4\:\text{cm}$
The area $(A)$ of an equilateral triangle,
$A = \frac{\sqrt{3}}{4}a^{2}= \frac{\sqrt{3}}{4}(4)^{2} = 4\sqrt{3}\:\text{cm}^{2}$
Hence, the correct answer is $4\sqrt{3}\:\text{cm}^{2}$.
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