Question

identify the correct phrase to complete the statement.
a central angle has a measure of 1 radian if the length of the intercepted arc is ______.
a. the same as the length of the diameter
b. 6.28 units
c. the same as the length of the radius
d. \\(\frac{\pi}{4}\\) units

Explanation:

Step1: Recall radian definition

The formula for arc length is \( s = r\theta \), where \( s \) is arc length, \( r \) is radius, and \( \theta \) is central angle in radians.

Step2: Apply \( \theta = 1 \) radian

Substitute \( \theta = 1 \) into \( s = r\theta \). We get \( s = r(1)=r \). So arc length equals radius when \( \theta = 1 \) radian.

Step3: Analyze options

• Option A: Diameter is \( 2r \), not equal to \( r \) (arc length for 1 radian). Eliminate.
• Option B: \( 6.28 \approx 2\pi \), not related to \( r \) for 1 radian. Eliminate.
• Option C: Matches \( s = r \) from Step2. Correct.
• Option D: \( \frac{\pi}{4} \) is a specific length, not related to \( r \) for 1 radian. Eliminate.

Answer:

C. the same as the length of the radius