Question

If $A,A_1,A_2$ and $A_3$and the areas of the incircle and excircles, then $\frac{1}{\sqrt{A_1}}+\frac{1}{\sqrt{A_2}}+\frac{1}{\sqrt{A_3}}$is equal to

• A. $1/\sqrt{A}$
• B. $2/\sqrt{A}$
• C. $3/\sqrt{A}$
• D. $4/\sqrt{A}$

Answer 1

The correct option is A

$1/\sqrt{A}$

The explanation for the correct option :

Step1. Assume the variables:

Let $r_1,r_2$and $r_3$ be the radius of the excircles.

Let the area of excircles

$$\begin{gathered} A_1=\pi {r_1}^2 \\ {A}_2=\pi {r_2}^2 \\ {A}_3=\pi {r_3}^2 \end{gathered}$$

Step2. Find the value of $\frac{1}{\sqrt{A_1}}+\frac{1}{\sqrt{A_2}}+\frac{1}{\sqrt{A_3}}$:

We have $\frac{1}{\sqrt{A_1}}+\frac{1}{\sqrt{A_2}}+\frac{1}{\sqrt{A_3}}=\frac{1}{\sqrt{\pi {r_1}^2}}+\frac{1}{\sqrt{\pi {r_2}^2}}+\frac{1}{\sqrt{\pi {r_3}^2}}$

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

$=\frac{1}{\sqrt{\pi }}\left(\frac{1}{r}\right)\left[\because \frac{\mathbf{1}}{{\mathbf{r}}_{\mathbf{1}}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{r}}_{\mathbf{2}}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{r}}_{\mathbf{3}}}\right]$

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

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

Hence, option (A) is the correct answer.