Circles
Inscribed Circle — Square Properties
nta_pyq_2024_apr
Grade 11
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:
Step-by-Step Solution
Key Concept: Inradius of equilateral triangle: $r=\frac{a}{2\sqrt{3}}=\frac{12}{2\sqrt{3}}=2\sqrt{3}$. Square inscribed in circle of radius $r$: side $=r\sqrt{2}=2\sqrt{6}$.
$r=2\sqrt{3}$, side of square $=2\sqrt{6}$. $m=24$, $n^2=384$. $m+n^2=408$.
Correct Answer: 1