The circle shown has center and circumference .
Which choice is the length of minor arc ?
When arc lengths (or central angles) are given as expressions like and , first identify what total they add up to by using the diagram (full circle, semicircle, or other fraction). Convert the given circumference to that total arc length, set up an addition equation (like total), solve for , and then compute the requested arc.
Hints
Find a half-circle
Look for the segment that passes through the center ; that segment indicates a diameter.
Turn circumference into semicircle length
A semicircle has half the circumference of the full circle.
Use the labeled arc expressions
The two labeled arcs on the semicircle add to the semicircle’s total arc length.
Desmos Guide
Compute the semicircle length
In Desmos, type 160*pi/2 to get the length of the semicircle arc from to .
Split the semicircle into 4 equal parts of
Type the result from step 1 divided by 4 to get the value of (since makes the semicircle).
Scale to get
Multiply the value of by 3. That result corresponds to the arc labeled , which is minor arc in the figure.
Step-by-step Explanation
Use the semicircle
Because is a diameter in the figure, the arc from to along the top is a semicircle.
So the length of arc (top semicircle) is half the circumference:
Use the arc labels to set a ratio
On that top semicircle, the arc from to is labeled , and the arc from to is labeled .
So their total is the semicircle:
Find the length of arc
Arc is labeled , so
Therefore, the correct answer is .