Question

In an equilateral triangle, the in-radius, circum-radius and one of the ex-radii are in the ratio

• A. \(2:3:5\)
• B. \(1:2:3\)
• C. \(1:3:7\)
• D. \(3:7:9\)

Hint:

For equilateral triangle, \(R = 2r\) and \(r_1 = 3r\).

Answer 1

The Correct Option is B

Step 1: Formula / Definition}
\[ r = \frac{\Delta}{s}, R = \frac{abc}{4\Delta}, r_1 = \frac{\Delta}{s-a} \]
Step 2: Calculation / Simplification}
For equilateral triangle of side \(a\):
\(\Delta = \frac{\sqrt{3}}{4}a^2, s = \frac{3a}{2}\)
\(r = \frac{\sqrt{3}a^2/4}{3a/2} = \frac{a}{2\sqrt{3}}\)
\(R = \frac{a^3}{4(\sqrt{3}a^2/4)} = \frac{a}{\sqrt{3}}\)
\(r_1 = \frac{\sqrt{3}a^2/4}{a/2} = \frac{\sqrt{3}a}{2}\)
\(r : R : r_1 = \frac{a}{2\sqrt{3}} : \frac{a}{\sqrt{3}} : \frac{\sqrt{3}a}{2} = 1 : 2 : 3\)
Step 3: Final Answer
\[ 1:2:3 \]