Library / /7 of 84

, 2014

Geometry Difficulty 4.7 AIME Prove it United States

Problem:

Let $O_{1}$ and $O_{2}$ be concentric circles with radii $4$ and $6$ , respectively. A chord $AB$ is drawn in $O_{1}$ with length $2$ . Extend $AB$ to intersect $O_{2}$ in points $C$ and $D$ . Find $CD$ .

Solution

Solution:

Let $O$ be the common center of $O_{1}$ and $O_{2}$ , and let $M$ be the midpoint of $AB$ . Then $OM \perp AB$ , so by the Pythagorean Theorem, $OM = \sqrt{4^{2} - 1^{2}} = \sqrt{15}$ . Thus $CD = 2CM = 2\sqrt{6^{2} - 15} = 2\sqrt{21}$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

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