Question 748·Medium·Math

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A circle has a circumference of $36\pi$ centimeters. Chord $PR$ subtends an inscribed angle $\angle PQR$ that measures $40^{\circ}$ . What is the length, in centimeters, of the minor arc $PR$ ?

An inscribed angle has its vertex on the circle and measures half the arc it intercepts. If the question asks for arc length, first double the inscribed angle to get the arc’s degree measure. Then take that fraction of $360^\circ$ and multiply it by the circumference. Using the given angle without doubling makes the arc too short.

Hints

Hint 1
An inscribed angle measures half the arc between the points where its sides meet the circle. To find the arc’s degree measure from the angle, should you halve or double?
Hint 2
A full circle measures $360^\circ$ . The arc takes up the same fraction of the circumference as its degree measure takes up of $360^\circ$ . Which angle belongs in that fraction?
Hint 3
The circumference is already the whole distance around the circle, so you don’t need to find the radius. Multiply the arc’s fraction by the given circumference.

Step-by-step

Double the angle, then use the circumference
Step 1
Find the arc’s degree measure
Because $\angle PQR$ is an inscribed angle, its vertex $Q$ is on the circle, and its sides meet the circle at $P$ and $R$ . An inscribed angle measures half the arc it intercepts. Double the given $40^\circ$ to find the arc’s degree measure:
$m\widehat{PR}=2(40^\circ)=80^\circ.$
Step 2
Find the arc’s share of the circle
The $80^\circ$ arc is less than half of $360^\circ$ , so it’s the minor arc. Its share of a full circle is:
$\frac{80}{360}=\frac{2}{9}.$
Don’t put $40^\circ$ in the numerator. That’s the inscribed angle, not the arc’s degree measure.
Step 3
Take that share of the circumference
The circle’s circumference is $36\pi$ centimeters. Type $\frac{80}{360}\cdot36$ in Desmos to find the coefficient of $\pi$ ; it shows $8$ . Keep the $\pi$ from the circumference. The minor arc length is $8\pi$ centimeters. Choice B.

Question 748·Medium·Math

Strategy

Hints

Step-by-step

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