# 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 $\stackrel{\Huge{\frown}}{ABC}$ is 16, pi.
What is the arc measure of $\stackrel{\LARGE{\frown}}{AC}$, in degrees?

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 $\stackrel{\Huge{\frown}}{ABC}$.
\begin{aligned}\text{Circumference} &= 2\pi r\\ &= 2\pi(10)\\ &= 20\pi \end{aligned}
We know the length of $\stackrel{\Huge{\frown}}{ABC}$, so we set up a proportion to figure out its arc measure.
\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 $\stackrel{\Huge{\frown}}{ABC}$ is 288, degree.
If we combine the major arc $\stackrel{\Huge{\frown}}{ABC}$ and the minor arc $\stackrel{\LARGE{\frown}}{AC}$, we have the entire circle.
\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 $\stackrel{\LARGE{\frown}}{AC}$ is 72, degree.

## Problem 2

In the figure below, the radius of circle P is 18 units. The arc length of $\stackrel{\LARGE{\frown}}{BA}$ is 14, pi.
What is the arc measure of $\stackrel{\LARGE{\frown}}{BC}$, in degrees?
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.
\begin{aligned}\text{Circumference} &= 2\pi r\\ &= 2\pi(18)\\ &= 36\pi \end{aligned}
We know the length of $\stackrel{\LARGE{\frown}}{BA}$, so we can set up a proportion to figure out its arc measure.
\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 $\stackrel{\LARGE{\frown}}{BC}$ and $\stackrel{\LARGE{\frown}}{CA}$ add up to the measure of $\stackrel{\LARGE{\frown}}{BA}$.
$\text{m}\stackrel{\LARGE{\frown}}{BC}+76^\circ = 140^\circ$
The arc measure of $\stackrel{\LARGE{\frown}}{BC}$ is 64, degree.