To determine the approximate measure, in radians, of the central angle corresponding to the arc [tex]\(AB\)[/tex] given that the ratio of the area of the sector [tex]\(AOB\)[/tex] to the area of the circle is [tex]\(\frac{3}{5}\)[/tex], we will follow these steps:
1. Understand the Ratio Given:
- The ratio of the area of the sector [tex]\(AOB\)[/tex] to the area of the entire circle is [tex]\(\frac{3}{5}\)[/tex].
2. Relate Area to Central Angle:
- In a circle, the area of a sector is proportional to its central angle. Specifically, the ratio of the area of the sector to the area of the circle is equivalent to the ratio of the central angle of the sector to the total angle of the circle (which is [tex]\(2\pi\)[/tex] radians for a complete circle).
3. Set Up the Equation:
- Given the ratio [tex]\(\frac{3}{5}\)[/tex], we equate this to the ratio of the central angle [tex]\(\theta\)[/tex] to [tex]\(2\pi\)[/tex]:
[tex]\[
\frac{\text{Area of sector}}{\text{Area of circle}} = \frac{\theta}{2\pi}
\][/tex]
[tex]\[
\frac{3}{5} = \frac{\theta}{2\pi}
\][/tex]
4. Solve for the Central Angle [tex]\(\theta\)[/tex]:
- Cross-multiply to solve for [tex]\(\theta\)[/tex]:
[tex]\[
\theta = \frac{3}{5} \times 2\pi
\][/tex]
Simplify this expression:
[tex]\[
\theta = \frac{6\pi}{5}
\][/tex]
To obtain a numeric value, compute:
[tex]\[
\theta \approx \frac{6 \times 3.14159}{5}
\][/tex]
[tex]\[
\theta \approx 3.7699111843077517
\][/tex]
5. Round to Two Decimal Places:
- Finally, round [tex]\(\theta\)[/tex] to two decimal places:
[tex]\[
\theta \approx 3.77
\][/tex]
Hence, the approximate measure, in radians, of the central angle corresponding to the arc [tex]\(AB\)[/tex] is [tex]\(\boxed{3.77}\)[/tex], which matches answer choice D.
