The side of an equilateral triangle is equal to the diagonal of the square. If the side of the square is 12 cm, then what is the area of the equilateral triangle?

Options

A. `50 sqrt""2`
B. `50 sqrt""3`
C. `72 sqrt""3`
D. `100 sqrt""3`

Answer

To solve the problem step by step, we will first find the diagonal of the square and then use that to calculate the area of the equilateral triangle.

### Step 1: Calculate the diagonal of the square
The formula for the diagonal \(d\) of a square with side length \(s\) is given by:
\[ d = s \sqrt{2} \]
Given that the side of the square \(s = 12 \, \text{cm}\), we can substitute this value into the formula:
\[ d = 12 \sqrt{2} \]

### Step 2: Determine the side length of the equilateral triangle
Since the side of the equilateral triangle is equal to the diagonal of the square, we have:
\[ \text{Side of the equilateral triangle} = 12 \sqrt{2} \, \text{cm} \]

### Step 3: Calculate the area of the equilateral triangle
The formula for the area \(A\) of an equilateral triangle with side length \(a\) is:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Substituting \(a = 12 \sqrt{2}\):
\[ A = \frac{\sqrt{3}}{4} (12 \sqrt{2})^2 \]

### Step 4: Simplify the expression
Calculating \((12 \sqrt{2})^2\):
\[ (12 \sqrt{2})^2 = 12^2 \cdot (\sqrt{2})^2 = 144 \cdot 2 = 288 \]
Now substituting back into the area formula:
\[ A = \frac{\sqrt{3}}{4} \cdot 288 \]

### Step 5: Final calculation
Now, simplifying:
\[ A = 72 \sqrt{3} \]

Thus, the area of the equilateral triangle is:
\[ \boxed{72 \sqrt{3}} \, \text{cm}^2 \]