Challenge problems: Arc length 2

Solve two challenging problems that ask you to find an arc measure using the arc length.

Problem 1

In the figure below, the radius of circle P is 10 units. The arc length of ABC\stackrel{\Huge{\frown}}{ABC} is 16, pi.
What is the arc measure of AC\stackrel{\LARGE{\frown}}{AC}, in degrees?
Choose 1 answer:
Choose 1 answer:

We are trying to figure out the arc measure of the arc that begins at point A and ends at point C.
We need to know the total circumference in order to determine the arc measure for ABC\stackrel{\Huge{\frown}}{ABC}.
Circumference=2πr=2π(10)=20π\begin{aligned}\text{Circumference} &= 2\pi r\\ &= 2\pi(10)\\ &= 20\pi \end{aligned}
We know the length of ABC\stackrel{\Huge{\frown}}{ABC}, so we set up a proportion to figure out its arc measure.
arc lengthcircumference=arc measuredegrees in a circle16π20π=arc measure360arc measure=36016π20π=288\begin{aligned} \dfrac{\text{arc length}}{\text{circumference}} &= \dfrac{\text{arc measure}}{\text{degrees in a circle}}\\\\\\ \dfrac{16\pi}{20\pi} &= \dfrac{\text{arc measure}}{360^\circ}\\\\\\ \text{arc measure} &=360^\circ \cdot\dfrac{16\pi}{20\pi}\\\\\\ &=288^\circ \end{aligned}
The arc measure of ABC\stackrel{\Huge{\frown}}{ABC} is 288, degree.
If we combine the major arc ABC\stackrel{\Huge{\frown}}{ABC} and the minor arc AC\stackrel{\LARGE{\frown}}{AC}, we have the entire circle.
288+mAC=360mAC=72\begin{aligned}288^\circ+ \text{m}\stackrel{\LARGE{\frown}}{AC}\, = 360^\circ\\ \text{m}\stackrel{\LARGE{\frown}}{AC}\, = 72^\circ\\\end{aligned}
The measure of AC\stackrel{\LARGE{\frown}}{AC} is 72, degree.

Problem 2

In the figure below, the radius of circle P is 18 units. The arc length of BA\stackrel{\LARGE{\frown}}{BA} is 14, pi.
What is the arc measure of BC\stackrel{\LARGE{\frown}}{BC}, in degrees?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i
degree

We are trying to figure out the arc measure of the arc that begins at point B and ends at point C.
Let's calculate the full circumference first, then solve proportionally for the arc measure based on the arc length.
Circumference=2πr=2π(18)=36π\begin{aligned}\text{Circumference} &= 2\pi r\\ &= 2\pi(18)\\ &= 36\pi \end{aligned}
We know the length of BA\stackrel{\LARGE{\frown}}{BA}, so we can set up a proportion to figure out its arc measure.
arc lengthcircumference=arc measuredegrees in a circle14π36π=arc measure360arc measure=14π36π360=140\begin{aligned} \dfrac{\text{arc length}}{\text{circumference}} &= \dfrac{\text{arc measure}}{\text{degrees in a circle}}\\\\\\ \dfrac{14\pi}{36\pi} &= \dfrac{\text{arc measure}}{360^\circ}\\\\\\ \text{arc measure} &=\dfrac{14\pi}{36\pi}\cdot {360^\circ}\\\\\\ &=140^\circ \end{aligned}
The measures of BC\stackrel{\LARGE{\frown}}{BC} and CA\stackrel{\LARGE{\frown}}{CA} add up to the measure of BA\stackrel{\LARGE{\frown}}{BA}.
mBC+76=140\text{m}\stackrel{\LARGE{\frown}}{BC}+76^\circ = 140^\circ
The arc measure of BC\stackrel{\LARGE{\frown}}{BC} is 64, degree.