Question 2·Medium·Circles

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A circle has center $O$ and circumference $120\pi$ . Points $A,B$ and $C$ lie on the circle such that the central angles $\angle AOB,\, \angle BOC$ , and $\angle COA$ are in the ratio $1:2:3$ . What is the length of arc $BC$ ?

For circle problems with central angles in a ratio, first convert the ratio into actual angle measures by adding the ratio numbers and dividing 360° by that sum. Use the specific ratio for the angle you care about to get its degree measure, then treat that angle as a fraction of the full circle (angle/360). Multiply this fraction by the total circumference to get the arc length—this avoids unnecessary work like finding the radius unless the problem specifically asks for it.

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Turn the ratio into actual angle measures

Add the numbers in the ratio $1:2:3$ to see how many equal parts the full circle (360°) is divided into, then find the degree measure of one part.

Focus on angle BOC

Once you know the size of one part, multiply by 2 to find the measure of $\angle BOC$ , since its ratio is 2.

Connect central angle and arc length

Use that arc length is a fraction of the full circumference, where the fraction is $\dfrac{\text{central angle}}{360^\circ}$ . Apply this with the central angle you found for $\angle BOC$ and the given circumference $120\pi$ .

Desmos Guide

Use Desmos to compute the arc length

In an expression line, type (120/360)*120*pi (or equivalently (1/3)*120*pi). Desmos will output a value in terms of $\pi$ ; that value is the length of arc $BC$ .

Step-by-step Explanation

Use the angle ratio to find one "part" of the circle

The central angles $\angle AOB$ , $\angle BOC$ , and $\angle COA$ are in the ratio $1:2:3$ .

Add the ratio numbers to find the total number of equal parts:

1 + 2 + 3 = 6

These 6 equal parts make up the full circle of $360^\circ$ , so each part has measure

\frac{360^\circ}{6} = 60^\circ.

Find the measure of angle BOC

The ratio for $\angle BOC$ is $2$ , meaning it is made of 2 of those equal parts.

So the measure of $\angle BOC$ is

2 \times 60^\circ = 120^\circ.

Relate the central angle to arc length

Arc length is proportional to its central angle. Specifically,

\text{arc length} = \frac{\text{central angle}}{360^\circ} \times \text{circumference}.

For arc $BC$ , the central angle is $120^\circ$ and the full circumference is $120\pi$ , so

\text{arc } BC = \frac{120^\circ}{360^\circ} \times 120\pi.

Compute the length of arc BC and choose the answer

Simplify the fraction first:

\frac{120}{360} = \frac{1}{3}.

\text{arc } BC = \frac{1}{3} \times 120\pi = 40\pi.

Therefore, the length of arc $BC$ is $40\pi$ , which corresponds to choice B.

Question 2·Medium·Circles

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation