Question

In any $\Delta ABC,$ minimum value of $\frac{r_1{r}_2{r}_3}{r^3}$ is equal to

• A. $3$
• B. $9$
• C. $27$
• D. None of these

Answer 1

Answer: (C)

$27$

$s=\frac{r_1{r}_2{r}_3}{r^3}$

$\frac{\frac{△}{s-a}\frac{△}{s-b}\frac{△}{s-c}}{\frac{{△}^3}{r^3}}$

$s=\frac{s^3}{(s-a)(s-b)(s-c)}$

$△=\sqrt{s(s-a)(s-b)(s-c)}$

$\frac{{△}^2}{s}=(s-a)(s-b)(s-c)$

$s=\frac{s^3}{\frac{{△}^2}{s}}\Rightarrow \frac{s^4}{{△}^2}$

$s$ will be minimum when $△$ is maximum

For given semi- perimeter, the area of a equilateral triangle is maximum,

$s=\frac{a+a+a}{2}=\frac{3a}{2}$

$s_{min}=\frac{(\frac{3a}{2}{)}^4}{(\frac{\sqrt{3}}{4}{a}^2{)}^2}=\frac{\frac{81}{16}}{\frac{3}{16}}$

$s_{min}=27$