. A square and an equilateral triangle have the same perimeter. If the diagonal of the square is 12√2 cm, then the area of the triangle is:a)64√2cm squareb)16√3cm squarec)64√3cm squared)62√3cm squaree)None of theseCorrect answer is option 'C'. Can you explain this answer?

Question

. A square and an equilateral triangle have the same perimeter. If the diagonal of the square is 12&radic;2 cm, then the area of the triangle is:a)64&radic;2cm squareb)16&radic;3cm squarec)64&radic;3cm squared)62&radic;3cm squaree)None of theseCorrect answer is option 'C'. Can you explain this answer?

Answer

Let's denote the side length of the square as $s$ and the side length of the equilateral triangle as $t$. We know that the perimeter of the square is equal to the perimeter of the triangle, so we can write the following equation:

$4s = 3t$

We also know that the diagonal of the square is equal to 12 units. The diagonal of a square divides it into two congruent right triangles, so we can use the Pythagorean theorem to find the relationship between the side length and the diagonal of the square:

$s^2 + s^2 = 12^2$

$2s^2 = 144$

$s^2 = 72$

$s = \sqrt{72} = 6\sqrt{2}$

Now, we can substitute the value of $s$ into the equation we found earlier:

$4(6\sqrt{2}) = 3t$

$24\sqrt{2} = 3t$

$t = \frac{24\sqrt{2}}{3} = 8\sqrt{2}$

Therefore, the side length of the equilateral triangle is $8\sqrt{2}$.

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