Points A and B lie on a circle with center O and radius 10. The measure of central angle AOB is 72^. The length of minor arc AB is k. What is the value of k?

Points A and B lie on a circle with center O and radius 10... | SAT Circles Question, Answer Explained

Math · Geometry and TrigonometryEasy

Points $A$ and $B$ lie on a circle with center $O$ and radius $10$ . The measure of central angle $AOB$ is $72^\circ$ . The length of minor arc $AB$ is $k\pi$ . What is the value of $k$ ?

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Answer: 4

The arc is $\frac{72}{360} = \frac{1}{5}$ of the circle. The circumference is $2\pi(10) = 20\pi$ , so the arc length is $\frac{1}{5}(20\pi) = 4\pi$ , and $k = 4$ .

Taking $\frac{1}{5}$ of the area, $100\pi$ , gives $20\pi$ , which is the area of the sector, not the length of the arc.

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