8. [25] A regular 12-sided polygon is inscribed in a circle of radius 1. How many chords of the circle that join two of the vertices of the 12-gon have lengths whose squares are rational? (No proof is necessary.)
Answer: 42
Solution: The chords joining vertices subtend minor arcs of $30^{\circ}, 60^{\circ}, 90^{\circ}, 120^{\circ}, 150^{\circ}$, or $180^{\circ}$. There are 12 chords of each of the first five kinds and 6 diameters. For a chord with central angle $\theta$, we can draw radii from the two endpoints of the chord to the center of the circle. By the law of cosines, the square of the length of the chord is $1+1-2 \cos \theta$, which is rational when $\theta$ is $60^{\circ}, 90^{\circ}, 120^{\circ}$, or $180^{\circ}$. The answer is thus $12+12+12+6=42$.