Question

which graph correctly represents the relationship between arc length and the measure of the corresponding central angle on a circle with radius r? a. graph with linear line from origin, y - axis labeled arc length, x - axis labeled radian measure of central angle with markings ( \frac{pi}{4} ), ( \frac{pi}{2} ), ( \frac{3pi}{4} ), ( 2pi )? (partial) b. partial graph

Explanation:

Step1: Recall the arc length formula

The formula for the arc length \( s \) of a circle with radius \( r \) and central angle \( \theta \) (in radians) is \( s = r\theta \). This is a linear equation in the form \( y = mx \) (where \( y = s \), \( x = \theta \), and \( m = r \)), which represents a straight line passing through the origin with a slope of \( r \).

Step2: Analyze the graphs

• For Option A: The graph is a straight line passing through the origin, which matches the linear relationship \( s = r\theta \).
• For Option B: The graph appears to be a curve (non - linear), which does not match the linear relationship between arc length and central angle (in radians) given by \( s = r\theta \).

Answer:

A