Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1971 Canadian MO Problems/Problem 1"

m (wikified)
m
Line 5: Line 5:
  
 
== Solution ==
 
== Solution ==
−
First, extend <math>\displaystyle CO</math> to meet the circle at <math>\displaystyle P.</math> Let the radius be <math>\displaystyle r.</math> Applying [[Power of a Point]],
+
First, extend <math>\displaystyle CO</math> to meet the circle at <math>\displaystyle P.</math> Let the radius be <math>\displaystyle r.</math> Applying [[power of a point]],
 
<math>\displaystyle (EP)(CE)=(BE)(ED)</math> and <math>\displaystyle 2r-1=15.</math> Hence, <math>\displaystyle r=8.</math>  
 
<math>\displaystyle (EP)(CE)=(BE)(ED)</math> and <math>\displaystyle 2r-1=15.</math> Hence, <math>\displaystyle r=8.</math>  
  

Revision as of 15:17, 26 July 2006

Problem

$\displaystyle DEB$ is a chord of a circle such that $\displaystyle DE=3$ and $\displaystyle EB=5 .$ Let $\displaystyle O$ be the center of the circle. Join $\displaystyle OE$ and extend $\displaystyle OE$ to cut the circle at $\displaystyle C.$ Given $\displaystyle EC=1,$ find the radius of the circle

CanadianMO 1971-1.jpg

Solution

First, extend $\displaystyle CO$ to meet the circle at $\displaystyle P.$ Let the radius be $\displaystyle r.$ Applying power of a point, $\displaystyle (EP)(CE)=(BE)(ED)$ and $\displaystyle 2r-1=15.$ Hence, $\displaystyle r=8.$

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1971_Canadian_MO_Problems/Problem_1&oldid=8504"