Question

Two arcs of the same circle have their lengths in the ratio 4:5. Find the ratio of the areas of the corresponding sectors.

Answer 1

Given:
Ratio of the two arcs of the circle = 4:5
Let the arcs be of lengths l₁ and l₂.
Now, according to question,
$\frac{l_1}{l_2}=\frac{4}{5}$ ...(i)
We know:
Area of the sector = $\frac{r}{2}\times \mathrm{Length}\mathrm{of}\mathrm{arc}$
Thus, we have:

$$\begin{gathered} \frac{{\mathrm{Area}}_1}{{\mathrm{Area}}_2}=\frac{{\displaystyle \frac{r}{2}}\times l_1}{\frac{r}{2}\times l_1} \\ =\frac{l_1}{l_2} \\ =\frac{4}{5} \end{gathered}$$ [Using (i)]

Thus, the ratio of the areas of the corresponding sectors is 4:5.