2021 Fall AMC 12B Problem 10

The three points lie on the unit circle at angles $40^\circ,$ $60^\circ,$ and $t^\circ.$ A chord’s length depends only on the angular separation of its endpoints. If the third point is equidistant from the other two, it lies on the perpendicular bisector: $t = 50$ or $t = 230.$ If its distance to $40^\circ$ equals the fixed chord (separation $20^\circ$), then $t = 20$ (since $t = 60$ is degenerate). If its distance to $60^\circ$ matches, then $t = 80$ (since $t = 40$ is degenerate). The valid values are $50, 230, 20, 80,$ summing to $380.$ Thus, the correct answer is E.