2021 AMC 12B Fall Problem 10

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),

and

\ \ (\cos t^\circ, \sin t^\circ)

is isosceles?

100

150

330

360

380

All three points lie on the unit circle, so equal sides mean equal chords, i.e. equal arc separations

Consider the three cases where the third point matches distance to

A,

to

B,

or lies on their perpendicular bisector

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.

Problem 10 in other years