The side of an equilateral triangle is equal to the diagonal of the square. If the side of the square is 10 cm, then what is the area of the equilateral triangle?

Options

A. a. `50 sqrt(2)`
B. b. `50 sqrt(3)`
C. c. `100 sqrt(2)`
D. d. `100 sqrt(3)`

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Find the diagonal of the square
The side length of the square is given as 10 cm. The diagonal \(d\) of a square can be calculated using the formula:
\[ d = s\sqrt{2} \]
where \(s\) is the side length of the square.

Substituting the value:
\[ d = 10\sqrt{2} \]

### Step 2: Set the side of the equilateral triangle equal to the diagonal
According to the problem, the side of the equilateral triangle \(A\) is equal to the diagonal of the square:
\[ A = 10\sqrt{2} \]

### Step 3: Calculate the area of the equilateral triangle
The area \(A_t\) of an equilateral triangle can be calculated using the formula:
\[ A_t = \frac{\sqrt{3}}{4} A^2 \]
Substituting the value of \(A\):
\[ A_t = \frac{\sqrt{3}}{4} (10\sqrt{2})^2 \]

### Step 4: Simplify the expression
Calculating \( (10\sqrt{2})^2 \):
\[ (10\sqrt{2})^2 = 100 \times 2 = 200 \]
Now substituting back into the area formula:
\[ A_t = \frac{\sqrt{3}}{4} \times 200 \]
\[ A_t = 50\sqrt{3} \]

### Step 5: Final result
Thus, the area of the equilateral triangle is:
\[ A_t = 50\sqrt{3} \text{ cm}^2 \]