Question

what is the ratio of the area of sector abc to the area of sector dbe?
a. $\frac{4}{3}$
b. $\frac{3}{4}$
c. $\frac{2}{3}$
d. $\frac{1}{3}$
e. $\frac{1}{4}$

Explanation:

Step1: Recall sector area formula

The area of a sector is $\frac{\theta}{360^\circ} \times \pi r^2$, where $\theta$ is the central angle and $r$ is the radius.

Step2: Define areas of both sectors

Let $A_{ABC}$ = area of sector $ABC$, $A_{DBE}$ = area of sector $DBE$.
$A_{ABC} = \frac{\beta}{360^\circ} \times \pi (2x)^2$, $A_{DBE} = \frac{2\beta}{360^\circ} \times \pi (x)^2$

Step3: Compute the ratio

$$\begin{align*} \frac{A_{ABC}}{A_{DBE}} &= \frac{\frac{\beta}{360^\circ} \times \pi (4x^2)}{\frac{2\beta}{360^\circ} \times \pi (x^2)} \\ &= \frac{4\beta x^2}{2\beta x^2} \\ &= \frac{4}{2 \times 2} \\ &= \frac{1}{4} \end{align*}$$

Answer:

E. $\frac{1}{4}$