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Cross the Circle (Posted on 2010-05-23) Difficulty: 2 of 5
Points A, B, and C are on a unit circle with A and C as end points of a diameter. The midpoint of chord AB is M. If the lengths of chord AB and line segment MC are the same, what is that length?

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Solution Solution | Comment 2 of 3 |

Draw segment BC.  Angle ABC is a right triangle because AC is a diameter.  Let AB=MC=x.  Then AM=MB=x/2.  Let diameter AC=1.  Let BC=y.

Then by Pythagorean Theorem:
(x/2)^2 + y^2 = x^2 from triangle MBC
x^2 + y^2 = 1^2 from triangle ABC

The first equation can be rewritten as y^2 = (3/4)x^2.  Substituting that into the second equation gives (7/4)x^2 = 1, or x = 2/sqrt(7) = 0.75593

  Posted by Brian Smith on 2010-05-23 18:53:28
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