If a circle is inscribed in an equilateral triangle of side a, then the area of any square (in sq. units)…

If a circle is inscribed in an equilateral triangle of side a, then the area of any square (in sq. units) inscribed in this circle is
  1. $\frac{2 a^2}{3}$
  2. $\sqrt{3} \frac{a^2}{2}$
  3. $\frac{a^2}{2 \sqrt{3}}$
  4. $\frac{a^2}{6}$

Solution


Area of $\triangle A B C(\Delta)=\frac{\sqrt{3}}{4} a^2$, Semi-perimeter of $\triangle A B C(s)=\frac{3 a}{2}$ Radius (r) $=\frac{\Delta}{s}=\frac{a}{2 \sqrt{3}}$ Diagonal of square PQRS $=2 \times \frac{a}{2 \sqrt{3}}=\frac{a}{\sqrt{3}}$ Area of square $=\frac{(\text { diagonal })^2}{2}=\frac{a^2}{6}$

Asked in: AP EAMCET 2024 (20 May Shift 1)

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