Question

a circular track has a radius of 100 meters. if a runner covers an arc length of 200 meters, what is the measure of the central angle in radians?
a. 2π radians
b. $\frac{pi}{2}$ radians
c. 2 radians
d. 1 radian

which of the following describes the radius of a circle?
a. the perimeter of the circle
b. the line segment from the center to any point on the circle
c. the line segment connecting two points on the circle
d. the line segment through the center connecting two points on the circle

which formula is used to find the area of a sector?
a. $a = pi r^{3}$
b. $a=\theta r$
c. $a = 2pi r$
d. $a=\frac{1}{2}\theta r^{2}$

Explanation:

Step1: Recall arc - length formula

The formula for arc - length $s$ of a circle is $s = r\theta$, where $r$ is the radius and $\theta$ is the central angle in radians.

Step2: Solve for the central angle

Given $s = 200$ meters and $r = 100$ meters. Rearranging the formula $\theta=\frac{s}{r}$, substituting the values we get $\theta=\frac{200}{100}=2$ radians.

Step3: Recall the definition of the radius

The radius of a circle is defined as the line segment from the center to any point on the circle.

Step4: Recall the sector - area formula

The formula for the area of a sector of a circle is $A=\frac{1}{2}\theta r^{2}$, where $\theta$ is the central angle in radians and $r$ is the radius.

Answer:

1. C. 2 radians
2. B. The line segment from the center to any point on the circle
3. D. $A = \frac{1}{2}\theta r^{2}$