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The height of an equilateral triangle is $\sqrt 6 cm$. Its area isA. $3\sqrt 3 c{m^2}$B. $2\sqrt 3 c{m^2}$C. $2\sqrt 2 c{m^2}$D. $6\sqrt 2 c{m^2}$

Last updated date: 13th Jun 2024
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Hint:
We shall first calculate the side of the equilateral triangle from its given height. Then using the side of the equilateral triangle, we will calculate its area.

Complete step by step solution:
We know that for an equilateral triangle of side $a$ and its height $h$ is given by,
$h = \dfrac{{\sqrt 3 }}{2}a$
The height given is $\sqrt 6$ cm. So the side of the triangle can be evaluated as
$\sqrt 6 = \dfrac{{\sqrt 3 }}{2}a \\ \Rightarrow a = 2\sqrt 2 \\$
The side of the triangle is 2 cm.
Now we know that the area of the equilateral triangle of side $a$ is given by
$Area = \dfrac{{\sqrt 3 }}{4}{a^2} \\ \Rightarrow Area = \dfrac{{\sqrt 3 }}{4}{(2\sqrt 2 )^2} \\ \Rightarrow Area = 2\sqrt 3 \\$
So, the area of the given triangle is $2\sqrt 3 c{m^2}$.

Therefore, the correct option is B.

Note:
An equilateral triangle has all its sides and angles equal. The perpendiculars from each vertex acts as the perpendicular bisectors of the opposite sides. The intersection of these perpendiculars (or altitudes) is known as an orthocenter. The length of an altitude of an equilateral triangle is also its height.