The circle above has center O, the length of arc `oversetfrown(ADC)` is `5pi` , and x=100. What is the length of arc `oversetfrown(ABC)` ?

Options

A. `9pi`
B. `13pi`
C. `18pi`
D. `13/2pi`

Answer

B

Explanation

The ratio of the lengths of two arcs of a circle is equal to the ratio of the measures of the central angles that subtend the arcs. It’s given that arc `oversetfrown(ADC)` is subtended by a central angle with measure `100^@`. Since the sum of the measures of the angles about a point is `360^@`, it follows that arc `oversetfrown(ABC)` is subtended by a central angle with measure `360^@ − 100^@ = 260^@`. If s is the length of arc `oversetfrown(ABC)` , then s must satisfy the ratio `s/(5pi) = 260/100` . Reducing the fraction `260/100` to its simplest form gives `13/5` . Therefore, `s/(5pi)` = `13/5` . Multiplying both sides of `s/(5pi) = 13/5` by `5pi` yields s = `13pi`.
Choice A is incorrect. This is the length of an arc consisting of exactly half of the circle, but arc `oversetfrown(ABC)` is greater than half of the circle. Choice C is incorrect. This is the total circumference of the circle. Choice D is incorrect. This is half the length of arc `oversetfrown(ABC)` , not its full length.