To find the measure of the angle of the sector, we can follow these steps:
1. Understand the Formula for the Area of a Sector:
The area [tex]\( A \)[/tex] of a sector is given by the formula:
[tex]\[
A = \frac{1}{2} r^2 \theta
\][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( \theta \)[/tex] is the central angle in radians.
2. Substitute the Given Values:
Since we know the area [tex]\( A = 52 \)[/tex] square inches and the radius [tex]\( r = 10 \)[/tex] inches, substitute these values into the formula:
[tex]\[
52 = \frac{1}{2} \times 10^2 \times \theta
\][/tex]
3. Solve for [tex]\( \theta \)[/tex] in Radians:
First, simplify:
[tex]\[
52 = \frac{1}{2} \times 100 \times \theta
\][/tex]
[tex]\[
52 = 50 \theta
\][/tex]
Now solve for [tex]\( \theta \)[/tex]:
[tex]\[
\theta = \frac{52}{50} = 1.04 \text{ radians}
\][/tex]
4. Convert Radians to Degrees:
To convert radians to degrees, use the conversion factor [tex]\( 180^\circ / \pi \)[/tex]:
[tex]\[
\theta \text{(in degrees)} = 1.04 \times \left( \frac{180}{\pi} \right)
\][/tex]
Approximating [tex]\( \pi \approx 3.14159 \)[/tex], we get:
[tex]\[
\theta \approx 1.04 \times 57.2958 = 59.5877^\circ
\][/tex]
5. Compare with Given Choices:
The given choices are:
- [tex]\( 84^\circ \)[/tex]
- [tex]\( 120^\circ \)[/tex]
- [tex]\( 60^\circ \)[/tex]
- [tex]\( 95^\circ \)[/tex]
The calculated angle [tex]\( 59.5877^\circ \)[/tex] is closest to [tex]\( 60^\circ \)[/tex].
Therefore, the measure of the angle of the sector is [tex]\( 60^\circ \)[/tex].
