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

Answer 1

Answer: (B)

We have; $\Delta=\frac{\sqrt{3}}{4} a^{2}, s=\frac{3a}{2}$
$\therefore r=\frac{\Delta}{S}=\frac{a}{2 \sqrt{3}}, R=\frac{a b c}{4 \Delta}$
$=\frac{a^{3}}{\sqrt{3} \cdot a^{2}}=\frac{a}{\sqrt{3}}$
and $r_{1}=\frac{\Delta}{S-a}=\frac{\sqrt{3} / 4 \cdot a^{2}}{a / 2}=\frac{\sqrt{3}}{2}\cdot a$
Hence; $r: R: r_{1}=\frac{a}{2 \sqrt{3}}: \frac{a}{\sqrt{3}}: \frac{\sqrt{3}}{2} \cdot a=1: 2: 3$