A circle is inscribed in a equilateral triangle of side 24 cm. What is the area ( in `cm^(2)` ) of a square inscribed in the circle ?

Options

A. 48
B. 72
C. 54
D. 96

Answer

To solve the problem step by step, we need to find the area of a square inscribed in a circle that is itself inscribed in an equilateral triangle with a side length of 24 cm.

### Step 1: Find the radius of the inscribed circle.
For an equilateral triangle, the radius \( r \) of the inscribed circle can be calculated using the formula:
\[ r = \frac{a}{2\sqrt{3}} \]
where \( a \) is the side length of the triangle.

Given \( a = 24 \) cm, we can substitute this value into the formula:
\[ r = \frac{24}{2\sqrt{3}} = \frac{12}{\sqrt{3}} \text{ cm} \]

### Step 2: Calculate the diameter of the inscribed circle.
The diameter \( D \) of the circle is twice the radius:
\[ D = 2r = 2 \times \frac{12}{\sqrt{3}} = \frac{24}{\sqrt{3}} \text{ cm} \]

### Step 3: Relate the diameter of the circle to the diagonal of the inscribed square.
The diagonal \( d \) of the square inscribed in the circle is equal to the diameter of the circle:
\[ d = D = \frac{24}{\sqrt{3}} \text{ cm} \]

### Step 4: Use the relationship between the diagonal and the side of the square.
The relationship between the diagonal \( d \) and the side \( s \) of the square is given by:
\[ d = s\sqrt{2} \]
Substituting the value of \( d \):
\[ \frac{24}{\sqrt{3}} = s\sqrt{2} \]

### Step 5: Solve for the side length \( s \) of the square.
To find \( s \), rearrange the equation:
\[ s = \frac{24}{\sqrt{3}\sqrt{2}} = \frac{24}{\sqrt{6}} \text{ cm} \]

### Step 6: Calculate the area of the square.
The area \( A \) of the square is given by:
\[ A = s^2 \]
Substituting the value of \( s \):
\[ A = \left(\frac{24}{\sqrt{6}}\right)^2 = \frac{576}{6} = 96 \text{ cm}^2 \]

Thus, the area of the square inscribed in the circle is \( 96 \text{ cm}^2 \).

### Summary of Steps:
1. Calculate the radius of the inscribed circle using the formula for an equilateral triangle.
2. Find the diameter of the circle.
3. Relate the diameter to the diagonal of the square.
4. Use the relationship between the diagonal and side of the square to find the side length.
5. Calculate the area of the square.