Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.7 AIME, harder Prove it Japan

Let $OAB$ be a sector of a circle with its center $O$ as in the figure below. A point $Q$ is chosen on the chord $AB$ . Let $P$ be the point of intersection of the arc of the circular sector and the line $OQ$ . If $AQ = 5$ , $BQ = 6$ , $OQ = PQ$ , determine the value of the radius of this sector. Here for a line segment $XY$ , its length is also denoted by $XY$ .

Solution

Let $r$ be the radius of the circle. Consider the full circle obtained by extending the arc of the circular sector in question. Let $R$ be the point of intersection, other than $P$ , of the line $PO$ and the full circle. By the well-known theorem on the power of a point with respect to a circle, we have $AQ \cdot BA = PQ \cdot RQ$ , which gives us

the equation $5 \cdot 6 = \frac{r}{2} \cdot \frac{3}{2} r$ . Solving this equation, we obtain $r = 2\sqrt{10}$ , which is the desired answer.

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