# 1973 AHSME Problems/Problem 15

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## Problem

A sector with acute central angle $\theta$ is cut from a circle of radius 6. The radius of the circle circumscribed about the sector is $\textbf{(A)}\ 3\cos\theta \qquad \textbf{(B)}\ 3\sec\theta \qquad \textbf{(C)}\ 3 \cos \frac12 \theta \qquad \textbf{(D)}\ 3 \sec \frac12 \theta \qquad \textbf{(E)}\ 3$

## Solution

Let $O$ be the center of the circle and $A,B$ be two points on the circle such that $\angle AOB = \theta$. If the circle circumscribes the sector, then the circle must circumscribe $\triangle AOB$. $[asy] draw((-120,-160)--(0,0)--(120,-160)); draw((-60,-80)--(0,-125)--(60,-80),dotted); draw((0,0)--(0,-125)); draw(arc((0,0),200,233.13,306.87)); [/asy]$

Draw the perpendicular bisectors of $OA$ and $OB$ and mark the intersection as point $C$, and draw a line from $C$ to $O$. By HL Congruency and CPCTC, $\angle AOC = \angle BOC = \theta /2$.

Let $R$ be the circumradius of the triangle. Using the definition of cosine for right triangles, $$\cos (\theta /2) = \frac{3}{R}$$ $$R = \frac{3}{\cos (\theta /2)}$$ $$R = 3 \sec (\theta /2)$$ Answer choices A, C, and E are smaller, so they are eliminated. However, as $\theta$ aproaches $90^\circ$, the value $3\sec\theta$ would approach infinity while $3\sec \tfrac12 \theta$ would approach $\tfrac{3\sqrt{2}}{2}$. A super large circle would definitely not be a circumcircle if $\theta$ is close to $90^\circ$, so we can confirm that the answer is $\boxed{\textbf{(D)}\ 3 \sec \frac12 \theta}$.

## See Also

 1973 AHSC (Problems • Answer Key • Resources) Preceded byProblem 14 Followed byProblem 16 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 All AHSME Problems and Solutions
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