A circle is inscribed in a square. An equilateral triangle of side `4sqrt(3) cm` is inscribed in that circle. The length of the diagonal of the square (in centimetres) is

Options

A. `4sqrt(2)`
B. 8
C. `8sqrt(2)`
D. 16

Answer

To find the length of the diagonal of the square in which a circle is inscribed, and an equilateral triangle of side \(4\sqrt{3}\) cm is inscribed in that circle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the relationship between the square, circle, and triangle:**
- Let the side length of the square be \(A\).
- The radius \(r\) of the inscribed circle is equal to half the side length of the square:
\[ r = \frac{A}{2} \]

2. **Use the formula for the radius of the circle inscribed in an equilateral triangle:**
- The radius \(r\) of the circle inscribed in an equilateral triangle can also be expressed as:
\[ r = \frac{s}{\sqrt{3}} \]
where \(s\) is the side length of the equilateral triangle.

3. **Substitute the known side length of the triangle:**
- Given that the side length of the triangle \(s = 4\sqrt{3}\) cm, we can substitute this into the formula:
\[ r = \frac{4\sqrt{3}}{\sqrt{3}} = 4 \text{ cm} \]

4. **Set the two expressions for the radius equal to each other:**
- From step 1 and step 3, we have:
\[ \frac{A}{2} = 4 \]

5. **Solve for \(A\):**
- Multiply both sides by 2:
\[ A = 8 \text{ cm} \]

6. **Calculate the diagonal of the square:**
- The formula for the diagonal \(d\) of a square is given by:
\[ d = A\sqrt{2} \]
- Substituting \(A = 8\):
\[ d = 8\sqrt{2} \text{ cm} \]

### Final Answer:
The length of the diagonal of the square is \(8\sqrt{2}\) cm.

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