Question

explain why the arc length, s, divided by the radius is the same, regardless of the circles radius. move words to the lines and expressions to the boxes to complete the statements. there are __ radians in a circle, so the ratio of the to the entire circle is given by the expression . the circumference of a circle with radius r is given by the expression . the arc length, s, is given by the expression . dividing the expression for arc length by r gives the expression . thus, the arc length divided by the radius is always equal to the measure of the __, regardless of the radius. 2π, πr², 2πr, θ, $\frac{\theta}{2pi}$, $\frac{\theta}{2pi r}$, 2πr×$\frac{\theta}{2pi}$, πr²×$\frac{\theta}{2pi r}$, area, arc length, central angle

Explanation:

Step1: Recall radian measure in a circle

There are $2\pi$ radians in a circle.

Step2: Write circumference formula

The circumference of a circle with radius $r$ is given by the expression $2\pi r$.

Step3: Find ratio of central - angle to full - circle

The ratio of the central angle $\theta$ to the entire circle is given by the expression $\frac{\theta}{2\pi}$.

Step4: Calculate arc - length formula

The arc length $s$ is given by the expression $2\pi r\times\frac{\theta}{2\pi}=\theta r$.

Step5: Divide arc - length by radius

Dividing the expression for arc length $s = \theta r$ by $r$ gives the expression $\theta$.

Answer:

There are $2\pi$ radians in a circle, so the ratio of the central angle to the entire circle is given by the expression $\frac{\theta}{2\pi}$. The circumference of a circle with radius $r$ is given by the expression $2\pi r$. The arc length, $s$, is given by the expression $2\pi r\times\frac{\theta}{2\pi}$. Dividing the expression for arc length by $r$ gives the expression $\theta$. Thus, the arc length divided by the radius is always equal to the measure of the central angle, regardless of the radius.