Question

A circle is inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are \( m \) and \( n \), respectively, then \( m + n^2 \) is equal to:

• A. 396
• B. 408
• C. 312
• D. 414

Answer 1

The Correct Option is B

The radius \( r \) of the circle inscribed in an equilateral triangle is given by:

\[ r = \frac{\Delta}{s} = \frac{\sqrt{3}a^2}{4a} = \frac{a}{2\sqrt{3}} = \frac{12}{2\sqrt{3}} = 2\sqrt{3}. \]

The side of the square inscribed in this circle is:

\[ \lambda = r\sqrt{2} = 2\sqrt{3} \cdot \sqrt{2} = 2\sqrt{6}. \]

Area of the square:

\[ m = \lambda^2 = (2\sqrt{6})^2 = 24. \]

Perimeter of the square:

\[ n = 4\lambda = 4(2\sqrt{6}) = 8\sqrt{6}. \]

\[ m + n^2 = 24 + (8\sqrt{6})^2 = 24 + 384 = 408. \]