What is the radius of the circle with centre at O_{2}?
Consider the following for the next three (03) items that follow: ABC is a right-angled triangle with \angle ABC=90^{\circ}. The centre of the incircle of the given triangle is at O, whose radius is 2 cm. Two more circles with centres at O_{1} and O_{2}, touch this circle and the two sides as shown in the figure given below. Further, MA:MC=2:3.
- A. 5-\sqrt{10}
- B. 1+2\sqrt{5}
- C. \frac{22-4\sqrt{10}}{9} ✓
- D. \frac{22-2\sqrt{10}}{9}
Correct Answer: C. \frac{22-4\sqrt{10}}{9}
Explanation
The distance CO = \sqrt{CM^2 + r^2} = \sqrt{6^2 + 2^2} = 2\sqrt{10}. For the circle O_2 in corner C, CO_2 = r_2\sqrt{10}. The total distance is CO = CO_2 + r_2 + r \implies 2\sqrt{10} = r_2\sqrt{10} + r_2 + 2. Solving for r_2 yields r_2 = \frac{2\sqrt{10}-2}{\sqrt{10}+1}, which rationalizes to \frac{22-4\sqrt{10}}{9}.
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