Question

in the diagram below, \\( \theta \\) is measured in radians. which equation represents the relationship between the radius, \\( r \\), and arc length, \\( s \\)?
diagram of a circle with center o, radius r, points a and b on the circumference, angle \\( \theta \\) at o, and arc s between a and b
\\( \bigcirc \\) \\( s = \theta \cdot r \\)
\\( \bigcirc \\) \\( r = \theta \cdot s \\)
\\( \bigcirc \\) \\( s = \theta + r \\)
\\( \bigcirc \\) \\( r = \theta + s \\)

Explanation:

Step1: Recall arc length formula

The formula for the length of an arc \( s \) of a circle with radius \( r \) and central angle \( \theta \) (in radians) is \( s = r\theta \) (or \( s=\theta\cdot r \)).

Step2: Analyze options

• Option 1: \( s = \theta\cdot r \) matches the arc length formula.
• Option 2: \( r=\theta\cdot s \) would imply \( r \) is proportional to \( s \) and \( \theta \), which is incorrect.
• Option 3: \( s=\theta + r \) is incorrect as arc length is a product, not sum, of \( \theta \) and \( r \).
• Option 4: \( r=\theta + s \) is also incorrect for the same reason as option 3.

Answer:

\( s = \theta \cdot r \) (the first option)