If a region bounded by a circle C is to be divided into three regions of equal areas by drawing two circles concentric with C, then the ratio of the radii of the two circles must be

### Concept & Equal Concentric Area Division When concentric circles divide a larger circle into regions of equal area, the areas of the consecutive circles are multiples of the smallest innermost circle's area. $$ \text{Area of circle} \propto (\text{Radius})^2 $$ ### Step-by-Step Solution * **Define regions:** Let the total area of circle $C$ be $A$. The three equal regions each have an area of $\frac{A}{3}$. * **Define the inner circles:** Let the smallest innermost circle have radius $r_1$. Its area is $\frac{A}{3}$. $\pi r_1^2 = \frac{A}{3}$ --- (Equation 1) * **Define the middle circle:** Let the next concentric circle have radius $r_2$. The area enclosed by this circle includes the innermost region and the middle region. Area enclosed by $r_2$ = $\frac{A}{3} + \frac{A}{3} = \frac{2A}{3}$. $\pi r_2^2 = \frac{2A}{3}$ --- (Equation 2) * **Find the ratio:** We need to find the ratio $r_1 : r_2$. Divide Equation 1 by Equation 2: $\frac{\pi r_1^2}{\pi r_2^2} = \frac{A/3}{2A/3}$ $\frac{r_1^2}{r_2^2} = \frac{1}{2}$ * **Take the square root:** $\frac{r_1}{r_2} = \frac{1}{\sqrt{2}} \Rightarrow r_1 : r_2 = 1 : \sqrt{2}$. ### Exam Strategy & Shortcut If a circle is divided into $n$ equal concentric area regions, the areas of the concentric circles from the center outwards are $1x, 2x, 3x, \dots, nx$. Since radius is proportional to the square root of the area, the radii will be in the ratio $\sqrt{1} : \sqrt{2} : \sqrt{3} : \dots : \sqrt{n}$. For the first two circles, the ratio is simply $\sqrt{1} : \sqrt{2}$ or $1 : \sqrt{2}$. ### Common Pitfall Students often confuse the area ratio with the linear ratio and incorrectly guess $1 : 2$ or $1 : 3$, forgetting to take the square root. ### Final Answer Therefore, the correct answer is **$1 : \sqrt{2}$**.