Competition problem — Geometry — Maths Olympiad Prep

Library / /33 of 94

Geometry Difficulty 4.6 AIME Prove it United States

Problem:

$A$ and $B$ are two points on a circle with center $O$ , and $C$ lies outside the circle, on ray $A B$ . Given that $A B = 24$ , $B C = 28$ , $O A = 15$ , find $O C$ .

Solution

Solution:

Let $M$ be the midpoint of $A B$ ; then $\triangle O M B$ is a right triangle with $O B = 15$ , $M B = 12$ , so $O M = 9$ .

Now $\triangle O M C$ is a right triangle with $O M = 9$ , $M C = 40$ , so $O C = 41$ .

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.