Question

Explain why the measure of the angle shown is the same regardless of the location in which the angle is measured. Use the definition θ = k s r in your explanation.

A) The measure of the angle is the same because it depends on the length of the arc (s) and the radius (r), not the location.
B) The measure of the angle is the same because it depends on the sine of the angle (θ) and the radius (r).
C) The measure of the angle is the same because it depends on the length of the hypotenuse (k) and the radius (r).
D) The measure of the angle is the same because it depends on the length of the side (s) and the radius (r).

Answer 1

The measure of an angle is consistent because it is determined by the ratio of the arc length to the radius of the circle. This ratio remains the same regardless of the location on the circle, ensuring that the angle measurement is constant.

Step-by-step explanation:

The measure of an angle in a circle is the same regardless of its location because it is determined by the relationship between the arc length (s) and the radius (r) of the circle. According to the definition θ = k s / r, the angle (θ) is calculated based on this relationship. The angle is measured in radians, which are dimensionless units derived from the ratio of the arc length to the radius. This means that as long as the ratio of the arc length to the radius remains constant, the angle will be the same, no matter where it is measured on the circle.

Therefore, the correct answer to why the measure of the angle is the same is that it depends on the length of the arc (s) and the radius (r), not on the location where the angle is measured, which corresponds to option A. The sine function, the length of the hypotenuse, or the length of the sides are not involved in the angle measurement in the context provided.