Question

A circle with radius r and arc s intercepted by central angle theta is shown. Question Explain why the arc length, s, divided by the radius is the same, regardless of the circle's radius. Move words to the lines and expressions to the boxes to complete the statements. Response area with 7 blank spaces There are Blank space 2 empty radians in a circle, so the ratio of the Blank space 3 empty to the entire circle is given by the expression Blank space 4 empty . The circumference of a circle with radius r is given by the expression Blank space 5 empty . The arc length, s, is given by the expression Blank space 6 empty . Dividing the expression for arc length by r gives the expression Blank space 7 empty . Thus, the arc length divided by the radius is always equal to the measure of the Blank space 8 empty , regardless of the radius. Answer options with 11 options. Answer Options 2 pi pi r squared 2 pi r theta StartFraction theta over 2 pi EndFraction

Answer 1

There are 2π radians in a circle, so the ratio of the angle θ to the entire circle is given by the expression θ/(2π). The circumference of a circle with radius r is given by the expression 2πr. The arc length s is given by the expression (θ/(2π))·(2πr) = θr. Dividing the expression for arc length by r gives the expression θ. Thus, the arc length divided by the radius is always equal to the measure of the central angle θ, regardless of the radius.