AMC 12 B

Geometry Difficulty 3.5 AMC 10/12 Find the answer United States

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?

Pick one

Solution

Let $A = (\cos 40^\circ, \sin 40^\circ)$ , $B = (\cos 60^\circ, \sin 60^\circ)$ , $C = (\cos t^\circ, \sin t^\circ)$ , and $O = (0, 0)$ . Then acute $\angle AOB = 20^\circ$ , so $\triangle ABC$ will be isosceles with vertex at $A$ or $B$ if $\angle AOC = 20^\circ$ or $\angle BOC = 20^\circ$ , respectively, which occurs when $t = 20$ or $t = 80$ .

The vertex will be $C$ if $\overrightarrow{OC}$ bisects either the acute or the reflex angle $AOB$ , which is $340^\circ$ . This will occur when $t = 50$ or $t = 230$ , respectively.

The requested sum is $20 + 80 + 50 + 230 = 380$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.