Question

The circle above with center O has a circumference of 36. What is the length of minor arc AC?

A) 9
B) 12
C) 18
D) 36

Answer 1

The length of minor arc AC is 9 units thus the correct option is A.

Step-by-step explanation:

The circumference of a circle is given by the formula
`\(C = 2\pi r\)`, where C represents the circumference and rndenotes the radius of the circle. In this case, the circumference is stated as 36 units. Using the formula for the circumference of a circle,
`\(36 = 2\pi r\).` To find the radius r, divide the circumference by
`\(2\pi\),` which gives
`\(r = (36)/(2\pi) = 18/\pi\)`.

The length of an arc is determined by the formula
`\(L = (n)/(360) * 2\pi r\)`, where L signifies the length of the arc, n is the angle subtended by the arc at the center of the circle (expressed in degrees), and r denotes the radius of the circle. Since the circumference of the circle is 36, and it's given that
`\(C = 2\pi r\)`, it follows that
`\(2\pi r = 36\),` implying that the radius r equals
`18/\(\pi\)`. Now, to calculate the length of minor arc AC, the central angle n formed by the arc needs to be known.

The ratio of the arc length to the circumference of the circle is proportional to the ratio of the angle subtended by the arc at the center to the total angle at the center (360 degrees for the complete circle).

Given that the circle's circumference is 36, which is one-fourth of the entire circumference, the angle subtended by the minor arc AC at the center is also one-fourth of the complete angle (360 degrees), resulting in
`\(n = 360 * (1)/(4) = 90\)` degrees. Using the formula for the arc length with the known angle and radius
`(\(L = (n)/(360) * 2\pi r\))`, substituting n = 90 degrees and
`\(r = 18/\pi\)` yields
`\(L = (90)/(360) * 2\pi * (18)/(\pi) = 9\)` units. Therefore, the length of minor arc AC is 9 units (option A).