2020/21 AHSMC Part 1 Problem 12

Problem 12 of 16

A right triangle has an angle of $60^\circ$ and the hypotenuse has length $2$ cm. An ant walks around the outside of the triangle always maintaining a distance of exactly $1$ cm to the nearest point on the triangle, and ending where it started. The distance (in cm) travelled by the ant is

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Solution

The sides of the triangle are of lengths $2$ , $1$ and $\sqrt{3}$ cm. The ant must trace the sides of the triangle and three arcs around each corner of lengths $\dfrac{\pi}{2}$ , $\dfrac{2\pi}{3}$ , $\dfrac{5\pi}{6}$ . Hence the ant travels a distance of $2 + 1 + \sqrt{3} + \frac{\pi}{2} + \frac{2\pi}{3} + \frac{5\pi}{6} = 3 + \sqrt{3} + 2\pi$ cm. The answer is (C).

University of Alberta

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