Circles
Circles Inscribed in Circles
Grade 11
Question:
<p>Consider 3 equal circles of radius \(r_1\) within a circle of radius \(r_2\) each to touch the other two and the given circle.</p><p><strong>Statement-1:</strong> \(\frac{r_1}{r_2} = \frac{\sqrt{3}}{\sqrt{3}+1}\)</p><p><strong>Statement-2:</strong> Incentre of triangle formed by joining centres of 3 equal circles is same as centre of given circle.</p>
<p>(A) Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for Statement-1</p>
<p>(B) Statement-1 is true, Statement-2 is true and Statement-2 is not correct explanation for Statement-1</p>
<p>(C) Statement-1 is true, Statement-2 is false</p>
<p>(D) Statement-1 is false, Statement-2 is true</p>
Step-by-Step Solution
Key Concept: The three circle centres form an equilateral triangle whose incentre coincides with the outer circle's centre due to symmetry.
<p>When 3 equal circles of radius \(r_1\) are inscribed in a circle of radius \(r_2\) such that each touches the other two and the outer circle, the centres form an equilateral triangle. The centre of the outer circle coincides with the incentre of this triangle. Using the geometry of the configuration, the distance from the centre to each inner circle's centre is \(r_2 - r_1\). For an equilateral triangle with side \(2(r_2-r_1)\), the circumradius equals \(r_2 - r_1\). This gives \(\frac{r_1}{r_2} = \frac{\sqrt{3}}{\sqrt{3}+1}\). Statement-2 correctly explains Statement-1.</p>
Correct Answer: A