Question

lf $A_1,A_2,A_3,A$(in-circle area) are respectively the areas of the ex-circles and in-circle of the $\Delta ABC$, then $\frac{1}{\sqrt{A_1}}+\frac{1}{\sqrt{A_2}}+\frac{1}{\sqrt{A_3}}=$

• A. $\frac{2}{\sqrt{A}}$
• B. $\frac{1}{\sqrt{A}}$
• C. $\frac{1}{2\sqrt{A}}$
• D. $\sqrt{A}$

Answer 1

The correct option is B $\frac{1}{\sqrt{A}}$

$A_1=\pi r_1^2,A_2=\pi r_2^2,A_3=\pi r_3^2,A=\pi r^2$

$=\frac{1}{\sqrt{\pi }}(\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3})$

$=\frac{1}{\sqrt{\pi }}\left(\frac{(s-a)+(s-b)+(s-c)}{\Delta }\right)$

$=\frac{1}{\sqrt{\pi }}\left(\frac{s}{\Delta }\right)$

$=\frac{1}{\sqrt{\pi }r}$

$=\frac{1}{\sqrt{\pi r^2}}$

$=\frac{1}{\sqrt{A}}$.