A square and an equilateral triangle have equal perimeters. If the diagonal of the square is $12\sqrt 2 $ cm, then area of the triangle is
${\text{A}}{\text{.}}$ $12\sqrt 2 c{m^2}$
${\text{B}}{\text{.}}$ $24\sqrt 3 c{m^2}$
${\text{C}}{\text{.}}$ $48\sqrt 3 c{m^2}$
${\text{D}}{\text{.}}$ $64\sqrt 3 c{m^2}$

Hint: find out the side of the square and triangle first and then put the value in the formula for finding out the area of the triangle.

Complete step-by-step answer:
Let us consider the side of the square as $x$ and triangle $y$, so we know that square has four sides and triangle has three sides…
So, for square $ = 4x$ and triangle $ = 3y$
Now, according to the question square and an equilateral triangle have equal perimeters,
$ \Rightarrow 4x = 3y \Rightarrow x = \dfrac{3}{4}y$
Diagonal of square $ = \sqrt 2 a$
$ \Rightarrow \sqrt 2 x = 12\sqrt 2 \Rightarrow x = 12cm \Rightarrow y = \dfrac{4}{3} \times 12 = 16cm$
Area of equilateral triangle $\begin{gathered}
   = \dfrac{{\sqrt 3 }}{4}{y^2} \Rightarrow \dfrac{{\sqrt 3 }}{4}\left( {256} \right) = 64\sqrt 3 \\
    \\
\end{gathered} $
So, option “D” is the correct answer.

Note: Area of a triangle is equal to half of the product of its base and height. The height of a triangle is the perpendicular distance from a vertex to the base of the triangle. In such questions, act strictly according to the question and then compare the possibilities to get an accurate answer.