A circle has center and circumference . Points and lie on the circle such that the central angles and are in the ratio . What is the length of arc ?

Solution available with step-by-step explanation

SAT Circles Question 2 | Free SAT Prep | Aniko

00:00

Question 2·Medium·Circles

0 / 50

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.

Hints

Click me!

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.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation