Key Concepts and Formulas

• Inradius of an Equilateral Triangle: The inradius ($r$) of an equilateral triangle with side length $S$ is given by $r = \frac{S}{2\sqrt{3}}$.
• Square Inscribed in a Circle: If a square with side length $a$ is inscribed in a circle of radius $R$, then the diagonal of the square equals the diameter of the circle, i.e., $a\sqrt{2} = 2R$, which implies $a = R\sqrt{2}$.
• Area and Perimeter of a Square: The area of a square with side $a$ is $a^2$, and its perimeter is $4a$.

Step-by-Step Solution

Step 1: Determine the Inradius of the Equilateral Triangle

We are given an equilateral triangle with side length 12, and we need to find the radius of the inscribed circle. This radius is the inradius of the triangle.

• Calculation: We use the formula for the inradius of an equilateral triangle: $r = \frac{S}{2\sqrt{3}}$ Substitute $S = 12$: $r = \frac{12}{2\sqrt{3}} = \frac{6}{\sqrt{3}}$ Rationalize the denominator: $r = \frac{6}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$ Therefore, the inradius (and the radius of the circle) is $2\sqrt{3}$.

Step 2: Find the Side Length of the Inscribed Square

Now we have a circle with radius $r = 2\sqrt{3}$, and a square is inscribed in it. We need to find the side length, $a$, of this square.

• Calculation: The diagonal of the square is equal to the diameter of the circle, so $a\sqrt{2} = 2r$. Solving for $a$: $a = \frac{2r}{\sqrt{2}} = \frac{2(2\sqrt{3})}{\sqrt{2}} = \frac{4\sqrt{3}}{\sqrt{2}}$ Rationalize the denominator: $a = \frac{4\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{6}}{2} = 2\sqrt{6}$ The side length of the inscribed square is $a = 2\sqrt{6}$.

Step 3: Calculate the Area ($m$) and Perimeter ($n$) of the Square

We have the side length of the square, $a = 2\sqrt{6}$. Now we calculate the area and perimeter.

• Calculation: Area of the square ($m$): $m = a^2 = (2\sqrt{6})^2 = 4 \cdot 6 = 24$

  Perimeter of the square ($n$): $n = 4a = 4(2\sqrt{6}) = 8\sqrt{6}$

Step 4: Compute the Final Expression $m+n^2$

We have $m = 24$ and $n = 8\sqrt{6}$. Now we compute $m + n^2$.

• Calculation: $n^2 = (8\sqrt{6})^2 = 64 \cdot 6 = 384$ $m + n^2 = 24 + 384 = 408$

Common Mistakes & Tips

• Rationalizing Denominators: Always rationalize denominators to simplify expressions and avoid errors.
• Using the Correct Formula: Ensure you are using the correct formula for the inradius of an equilateral triangle.
• Squaring Radicals: Remember to square both the coefficient and the radical when squaring an expression like $8\sqrt{6}$.

Summary

We first found the inradius of the equilateral triangle, which is also the radius of the circle. Then, we used the relationship between the side length of an inscribed square and the radius of the circle to find the side length of the square. Finally, we calculated the area and perimeter of the square and computed the value of the expression $m+n^2$.

The final answer is $\boxed{408}$, which corresponds to option (A).