The side of an equilatetral triangle has the same length as the diagonal of a square. What is the area of the square?
(1) The height of the equilateral triangle is equal to `6sqrt(3)`.
(2) The area of the equilateral triangle is equal to `36sqrt(3)`.

Answer

To solve the problem step by step, we need to find the area of the square given that the side of an equilateral triangle has the same length as the diagonal of the square. We will use the information provided in the statements.

### Step 1: Understand the relationship between the equilateral triangle and the square
Let the side length of the equilateral triangle be \( a \). The diagonal \( D \) of the square is equal to the side length of the triangle, so we have:
\[ D = a \]

### Step 2: Use the height of the equilateral triangle
We know that the height \( h \) of an equilateral triangle can be expressed in terms of its side length \( a \) as:
\[ h = \frac{\sqrt{3}}{2} a \]
Given that the height \( h \) is equal to \( 6\sqrt{3} \), we can set up the equation:
\[ \frac{\sqrt{3}}{2} a = 6\sqrt{3} \]

### Step 3: Solve for \( a \)
To find \( a \), we can multiply both sides of the equation by \( \frac{2}{\sqrt{3}} \):
\[ a = 6\sqrt{3} \cdot \frac{2}{\sqrt{3}} = 12 \]

### Step 4: Find the diagonal of the square
Since the diagonal \( D \) of the square is equal to the side length \( a \) of the equilateral triangle, we have:
\[ D = 12 \]

### Step 5: Calculate the area of the square
The area \( A \) of a square in terms of its diagonal \( D \) is given by the formula:
\[ A = \frac{D^2}{2} \]
Substituting \( D = 12 \):
\[ A = \frac{12^2}{2} = \frac{144}{2} = 72 \]

### Final Answer
The area of the square is \( 72 \) square units.

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