To solve this problem, follow these steps:
1. Understand the relationship: The problem states that the ratio of the area of sector [tex]\( AOB \)[/tex] to the area of the entire circle is [tex]\( \frac{3}{5} \)[/tex].
2. Formula for the area of a circle: Recall that the area of a circle is given by [tex]\( \pi r^2 \)[/tex], where [tex]\( r \)[/tex] is the radius of the circle.
3. Formula for the area of a sector: The area of a sector of a circle with central angle [tex]\( \theta \)[/tex] (in radians) is given by [tex]\( \frac{\theta}{2\pi} \cdot \pi r^2 \)[/tex].
4. Set up the equation: According to the given ratio [tex]\( \frac{3}{5} \)[/tex], we can write:
[tex]\[
\frac{\text{Area of sector } AOB}{\text{Area of circle}} = \frac{3}{5}
\][/tex]
Substituting the formulas for the areas, we get:
[tex]\[
\frac{\frac{\theta}{2\pi} \cdot \pi r^2}{\pi r^2} = \frac{3}{5}
\][/tex]
5. Simplify the equation: The [tex]\( \pi r^2 \)[/tex] terms cancel out, leaving us with:
[tex]\[
\frac{\theta}{2\pi} = \frac{3}{5}
\][/tex]
6. Solve for [tex]\( \theta \)[/tex]: To isolate [tex]\( \theta \)[/tex], multiply both sides by [tex]\( 2\pi \)[/tex]:
[tex]\[
\theta = \frac{3}{5} \cdot 2\pi
\][/tex]
7. Calculate the value of [tex]\( \theta \)[/tex]:
[tex]\[
\theta = \frac{3}{5} \cdot 2\pi \approx 3.7699111843077517
\][/tex]
8. Round the answer: Round the value of [tex]\( \theta \)[/tex] to two decimal places:
[tex]\[
\theta \approx 3.77
\][/tex]
The approximate measure of the central angle corresponding to [tex]\( \widehat{AB} \)[/tex], rounded to two decimal places, is [tex]\( 3.77 \)[/tex] radians.
Thus, the correct answer is:
[tex]\[
\boxed{3.77}
\][/tex]
