To find the degree measure of the angle subtended at the center of a circle by an arc, we need to use the relationship between the arc length, radius, and the angle subtended. Here's the step-by-step solution:
1. Given Values:
- Diameter of the circle: [tex]\(200\)[/tex] cm
- Arc length: [tex]\(22\)[/tex] cm
- [tex]\(\pi \approx \frac{22}{7}\)[/tex]
2. Calculate the Radius:
The radius is half of the diameter. Therefore,
[tex]\[
\text{Radius} = \frac{\text{Diameter}}{2} = \frac{200}{2} = 100 \text{ cm}
\][/tex]
3. Calculate the Angle in Radians:
The angle subtended by an arc at the center of a circle (in radians) is given by the formula:
[tex]\[
\text{Angle (radians)} = \frac{\text{Arc Length}}{\text{Radius}}
\][/tex]
Substituting the given values:
[tex]\[
\text{Angle (radians)} = \frac{22}{100} = 0.22 \text{ radians}
\][/tex]
4. Convert the Angle from Radians to Degrees:
To convert from radians to degrees, we use the conversion factor that [tex]\(180\)[/tex] degrees is equal to [tex]\(\pi\)[/tex] radians. Therefore,
[tex]\[
\text{Angle (degrees)} = \text{Angle (radians)} \times \frac{180}{\pi}
\][/tex]
Substituting the values:
[tex]\[
\text{Angle (degrees)} = 0.22 \times \frac{180}{\left(\frac{22}{7}\right)} = 0.22 \times \frac{180 \times 7}{22} = 0.22 \times \frac{1260}{22} = 0.22 \times 57.27 \approx 12.6 \text{ degrees}
\][/tex]
Therefore, the degree measure of the angle subtended at the center of the circle by an arc of length 22 cm is approximately [tex]\(12.6\)[/tex] degrees.
