Question

explore radian measure by following these steps. 1. move point k to set m∠jpk to 1 radian. for both circles, what is the relationship between the arc length and radius? arc length < radius arc length = radius arc length > radius m∠jpk = 1.00 radians inner circle pg = 1.00 units length of gh = 1.00 units outer circle length of jk = 3.00 units pj = 3.00 units

Explanation:

Step1: Recall the radian formula

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

Step2: Analyze the inner circle

For the inner circle, the radius \( PG = 1.00 \) units and the central angle \( \theta = 1.00 \) radians. Using the arc - length formula \( s=r\theta \), we substitute \( r = 1.00 \) and \( \theta=1.00 \). So \( s = 1.00\times1.00=1.00 \) units. The arc length \( GH = 1.00 \) units and the radius \( PG = 1.00 \) units, so \( GH=PG \).

Step3: Analyze the outer circle

For the outer circle, the radius \( PJ = 3.00 \) units and the central angle \( \theta = 1.00 \) radians. Using the arc - length formula \( s = r\theta \), we substitute \( r = 3.00 \) and \( \theta = 1.00 \). So \( s=3.00\times1.00 = 3.00 \) units. The arc length \( JK = 3.00 \) units and the radius \( PJ = 3.00 \) units, so \( JK = PJ \).

From the analysis of both circles, when the central angle is 1 radian, the arc length is equal to the radius.

Answer:

arc length = radius