Question

a circle with radius r and arc s intercepted by central angle θ is shown. 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, θ/2π, θ/2πr, 2πr×θ/2π, πr²×θ/2πr, area, arc length, central angle

Explanation:

Step1: Recall the total radians in a circle

There are \(2\pi\) radians in a circle.

Step2: Find the ratio of the central angle to the entire circle

The ratio of the central angle \(\theta\) to the entire circle is \(\frac{\theta}{2\pi}\).

Step3: Recall the formula for circumference

The circumference of a circle with radius \(r\) is \(2\pi r\).

Step4: Calculate the arc length formula

The arc length \(s\) is \(2\pi r\times\frac{\theta}{2\pi}\) (since arc length is a fraction of the circumference based on the central - angle ratio).

Step5: Simplify the arc - length - divided - by - radius formula

Dividing the arc - length formula \(s = 2\pi r\times\frac{\theta}{2\pi}\) by \(r\): \(\frac{2\pi r\times\frac{\theta}{2\pi}}{r}=\theta\).

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.