Question : The height of an equilateral triangle is $9 \sqrt{3} \mathrm{~cm}$. What is the area of this equilateral triangle?
Option 1: $92 \sqrt{3} \mathrm{~cm}^2$
Option 2: $67 \sqrt{3} \mathrm{~cm}^2$
Option 3: $49 \sqrt{3} \mathrm{~cm}^2$
Option 4: $81 \sqrt{3} \mathrm{~cm}^2$
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Correct Answer: $81 \sqrt{3} \mathrm{~cm}^2$
Solution :
Let the sides of the equilateral triangle be $a$, then its height = $\frac{a\sqrt3}{2}$
⇒ $9\sqrt3 = \frac{a\sqrt{3}}{2}$
$\therefore a = 18$ cm
Now, the area of the triangle
= $\frac{a^2\sqrt3}{4}$
= $\frac{18×18×\sqrt3}{4}$
= $81 \sqrt{3} \mathrm{~cm}^2$
Hence, the correct answer is $81 \sqrt{3} \mathrm{~cm}^2$.
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