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]] |
Latest revision as of 15:33, 4 September 2024
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 |