Question 764·Medium·Math

0 / 2000

Points $A$ , $B$ , $C$ , and $D$ lie in that order on a circle with center $O$ . The measures of arcs $AB$ , $BC$ , and $CD$ are $40^\circ$ , $70^\circ$ , and $110^\circ$ , respectively. What is the radian measure of $\angle AOC$ ?

An angle with its vertex at the circle’s center is a central angle. Its degree measure equals the measure of the arc between its endpoints. Use the order of the points to add only the arcs on that path, then convert degrees to radians using $180^\circ=\pi$ radians. Don’t halve the arc; that rule applies to an angle whose vertex is on the circle.

Hints

Hint 1
An arc is a portion of the circle. To travel from $A$ to $C$ in the order given, you pass through $B$ . Which two arcs make up that path?
Hint 2
A central angle has its vertex at the center and matches its arc in degrees. Once you have that degree measure, use $180^\circ=\pi$ radians to change units, not the size of the angle.

Step-by-step

Follow the arc, then convert
Step 1
Find the arc from A to C
The points lie in order $A$ , $B$ , $C$ , $D$ , so the path from $A$ to $C$ through $B$ contains arcs $AB$ and $BC$ . Add only those two arcs: $m\widehat{AC}=40^\circ+70^\circ=110^\circ$ . Arc $CD$ comes after $C$ , so it isn’t part of this path.
Step 2
Match the central angle to its arc
$O$ is the center, so $\angle AOC$ is a central angle. A central angle has the same degree measure as the arc it intercepts. Its sides reach the circle at $A$ and $C$ , so $m\angle AOC=110^\circ$ . Don’t halve it: that would be the rule for an angle with its vertex on the circle.
Step 3
Convert the angle to radians
Since $180^\circ=\pi$ radians, the angle’s coefficient of $\pi$ is $110/180$ . Type $110/180$ in Desmos. It shows $0.611111\ldots$ ; click the fraction button to get $11/18$ . Keep $\pi$ in the answer: $\angle AOC$ measures $\frac{11\pi}{18}$ radians. Choice D.

Question 764·Medium·Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this