Circle 3 Question 18
18. A circle is inscribed in an equilateral triangle of side $a$. The area of any square inscribed in this circle is… .
(1994, 2M)
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Answer:
Correct Answer: 18. $\frac{a^{2}}{6}$ sq unit
Solution:
- In an equilateral triangle, the radius of incircle
$$ \begin{aligned} & =\frac{1}{3} \times \text { median of the triangle } \\ & =\frac{1}{3} \sqrt{a^{2}-\frac{a^{2}}{4}}=\frac{1}{3} \sqrt{\frac{4 a^{2}-a^{2}}{4}}=\frac{a}{2 \sqrt{3}} \end{aligned} $$
Therefore, area of the square inscribed in this circle
$$ =2(\text { radius of circle })^{2}=\frac{2 a^{2}}{4 \cdot 3}=\frac{a^{2}}{6} \text { sq unit } $$
