In $\triangle ABC $, $AB=10$, $AC=18$, and $\angle B=130^\circ.$ Let $O$ be the center of the circle containing $A$, $B$, and $C.$ What is the degree measure of $\angle CAO$?
$$\textbf{(A) } 20 \qquad \textbf{(B) } 30 \qquad \textbf{(C) } 40 \qquad \textbf{(D) } 50 \qquad \textbf{(E) } 60$$
Difficulty: 31/100 — Reliant on knowledge of inscribed angle theorem, and can be tricky initially.
Core Concepts: circles, inscribed angles, circumcenter
Challenges: Without familiarity with the inscribed angle theorem, the problem is quite difficulty to approach. Even then, it may not be apparent to determine how to get the arc measure desired.
Techniques: Labeling the measures of each arc to determine the measure of the angle it subtends.
Error-prone Steps: Assuming $O$ is in the triangle, arithmetic errors, or foregtting to divide by $2$ in the inscribed angle theorem.
Ideal Time:
Experienced: ≤ 30 sec
Intermediate: ≤ 45-90 sec
Beginner: ≤ 3 min