Points A and B lie on a circle with center O and radius 10. The length of chord AB is 103. What is the length of the minor arc AB?

SAT Circles Practice Question 3 (Hard)

Math · Geometry and TrigonometryHard

Points $A$ and $B$ lie on a circle with center $O$ and radius $10$ . The length of chord $\overline{AB}$ is $10\sqrt{3}$ . What is the length of the minor arc $AB$ ?

$\frac{10\pi}{3}$

$5\pi$

$\frac{40\pi}{3}$

$\frac{20\pi}{3}$

Show answer and explanation

Answer: D

Draw the perpendicular from $O$ to the chord; it bisects $\overline{AB}$ into halves of length $5\sqrt{3}$ . In the resulting right triangle with hypotenuse $10$ , $\sin(\angle AOM) = \frac{5\sqrt{3}}{10} = \frac{\sqrt{3}}{2}$ , so $\angle AOM = 60^\circ$ and $\angle AOB = 120^\circ$ . The minor arc length is $\frac{120}{360} \cdot 2\pi(10) = \frac{20\pi}{3}$ .

Choice A results from taking the central angle to be $60^\circ$ , which is only half of angle $AOB$ .

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