Question

What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are $(\cos 40^\circ, \sin 40^\circ),\ (\cos 60^\circ, \sin 60^\circ),\ \text{and}\ (\cos t^\circ, \sin t^\circ)$ is isosceles?

$100$

$150$

$330$

$360$

$380$

Solution:

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.

2021 AMC 12B Fall Problem 10

Solution:

Problem 10 in Other Years