Q70 / 160 · TS EAMCET 2021 (04 Aug Shift 2) · Mathematics

In an equilateral triangle $A B C, r : R : r_{1}$ is

Solution

In an Equilateral triangle, $a = b = c$

$∴ s = \frac{a + b + c}{2} = \frac{3 a}{2}$ & $∆ = \frac{\sqrt{3}}{4} a^{2}$

$∴ r = \frac{∆}{s} = \frac{∆}{\frac{3 a}{2}} = \frac{2 ∆}{3 a} = \frac{a}{2 \sqrt{3}}$

Also, $R = \frac{a b c}{4 ∆} = \frac{a^{3}}{4 \frac{\sqrt{3}}{4} a^{2}} = \frac{a}{\sqrt{3}}$

And $r_{1} = \frac{∆}{s - a} = \frac{∆}{\frac{3 a}{2} - a} = \frac{2 ∆}{a} = \frac{\sqrt{3}}{2} a$

Now, $r : R : r_{1} = \frac{a}{2 \sqrt{3}} : \frac{a}{\sqrt{3}} : \frac{\sqrt{3} a}{2} = 1 : 2 : 3$

Hence, the correct answer is option: 'C'.