Question 41·Medium·Circles

As you travel around the circle in order $A \to B \to C \to D$, which smaller arcs do you pass through when going from $A$ to $C$ without going past $C$?

Once you find the degree measure of arc $AC$, remember that to convert degrees to radians you multiply by $\frac{\pi}{180}$ and then simplify the fraction.

In a Desmos expression line, type 40 + 70 to confirm that the central angle $AOC$ (which intercepts arc $AC = AB + BC$) has measure $110$ degrees.

In a new Desmos line, type (40 + 70)*pi/180 or 110*pi/180. Desmos will show this value in terms of $\pi$; match that simplified expression to the correct answer choice.

Angle $AOC$ is a central angle (its vertex is at the center $O$), so its measure equals the measure of the arc it intercepts.

Because the points lie in order $A$, $B$, $C$, $D$ around the circle, the arc from $A$ to $C$ that does not pass through $D$ is made of arcs $AB$ and $BC$ together. So $\angle AOC$ intercepts arc $AC = \text{arc }AB + \text{arc }BC$.

Add the given arc measures for $AB$ and $BC$:

Since $\angle AOC$ is a central angle intercepting arc $AC$, $m\angle AOC = 110^\circ$.

Use the conversion formula from degrees to radians:

Substitute $110^\circ$:

Now simplify the fraction $\frac{110}{180}$ by dividing numerator and denominator by $10$ first, then by $2$:

So the radian measure of $\angle AOC$ is