To find the measure of the central angle whose radii define the arc on a circle, you can use the formula:
\[ \text{Central Angle} = \left( \frac{\text{Arc Length}}{\text{Circle Circumference}} \right) \times 360^\circ \]
Given:
- Radius of the circle = 10 centimeters
- Arc length = 8 centimeters
First, calculate the circle's circumference using the formula:
\[ \text{Circle Circumference} = 2 \times \pi \times \text{Radius} \]
Substitute the radius value:
\[ \text{Circle Circumference} = 2 \times \pi \times 10 = 20\pi \]
Now, substitute the arc length and circle circumference into the central angle formula:
\[ \text{Central Angle} = \left( \frac{8}{20\pi} \right) \times 360^\circ \]
Simplify the fraction:
\[ \text{Central Angle} = \left( \frac{2}{5\pi} \right) \times 360^\circ \]
To find the answer, calculate the value:
\[ \text{Central Angle} ≈ 72.57° \]
Rounded to the nearest whole number, the measure of the central angle whose radii define the arc is approximately 73 degrees. Therefore, the closest option among the choices provided is 72 degrees.
