Difference between revisions of "1971 Canadian MO Problems/Problem 1"
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== Problem == | == Problem == | ||
− | <math> | + | <math>DEB</math> is a chord of a circle such that <math>DE=3</math> and <math>EB=5 .</math> Let <math>O</math> be the center of the circle. Join <math>OE</math> and extend <math>OE</math> to cut the circle at <math>C.</math> Given <math>EC=1,</math> find the radius of the circle. |
[[Image:CanadianMO_1971-1.jpg]] | [[Image:CanadianMO_1971-1.jpg]] | ||
== Solution == | == Solution == | ||
− | First, extend <math> | + | First, extend <math>CO</math> to meet the circle at <math>P.</math> Let the radius be <math>r.</math> Applying [[power of a point]], |
− | <math> | + | <math>(EP)(CE)=(BE)(ED)</math> and <math>2r-1=15.</math> Hence, <math>r=8.</math> |
− | - | + | == See Also == |
− | + | {{Old CanadaMO box|before=First Question|num-a=2|year=1971}} | |
− | |||
[[Category:Intermediate Geometry Problems]] | [[Category:Intermediate Geometry Problems]] |
Latest revision as of 20:38, 9 July 2019
Problem
is a chord of a circle such that and Let be the center of the circle. Join and extend to cut the circle at Given find the radius of the circle.
Solution
First, extend to meet the circle at Let the radius be Applying power of a point, and Hence,
See Also
1971 Canadian MO (Problems) | ||
Preceded by First Question |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • | Followed by Problem 2 |