CircleHardBloom L3
Question
In the $xy$-plane, the length of the shortest path from $(0,0)$ to $(12,16)$ that does not go inside the circle $(x - 6)^{2} + (y - 8)^{2} = 25$ is:
Options
A.$10\sqrt{3}$
B.$10\sqrt{5}$
C.$10\sqrt{3} + \frac{5\pi}{3}$
D.$10 + 5\pi$
Solution
[Key Concept: Shortest path avoiding a circle = two tangent segments + the arc (unfold the geometry).]
$OABC$ in the shortest path
$$\begin{matrix} OA + \overline{AB} + BC & \ = 10cos30^{\circ} + \frac{5\pi}{3} + 10cos30^{\circ} \\ & \ = 10\sqrt{3} + \frac{5\pi}{3} \end{matrix}$$
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