What is the radius of the circle with centre at O_{1}?
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. 4-\sqrt{5}
- B. 1+\sqrt{5}
- C. 2+\sqrt{5}
- D. 3-\sqrt{5} ✓
Correct Answer: D. 3-\sqrt{5}
Explanation
The distance AO = \sqrt{AM^2 + r^2} = \sqrt{4^2 + 2^2} = 2\sqrt{5}. For the smaller circle O_1 nestled in corner A, let its radius be r_1. Due to similar triangles, AO_1 = r_1\sqrt{5}. We know AO = AO_1 + r_1 + r \implies 2\sqrt{5} = r_1\sqrt{5} + r_1 + 2. Solving for r_1 gives \frac{2\sqrt{5}-2}{\sqrt{5}+1}, which rationalizes to 3-\sqrt{5}.
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