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Circle Line Length (Posted on 2016-04-07) Difficulty: 2 of 5
Each of the points P and Q lie on a circle with its center at O and having a radius of √50. Point R is inside the circle such that:
∠PQR = 90o, PQ = 6, QR = 2
Find OR

No Solution Yet Submitted by K Sengupta    
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Solution Solution | Comment 1 of 3
Let S be the opposite endpoint of the diameter containing P.  The angle PQS is then a right angle, which means that R is on QS.

By Pythagoras QS = 2*sqrt(41)  Then RS = 2*sqrt(41)-2

Angle PSQ is the same as angle OSR, and cos(angle PSQ) = sqrt(41/50)

Then by the law of cosines OR^2 = 50 + (2*sqrt(41)-2)^2 - 2*sqrt(50)*(2*sqrt(41)-2)*sqrt(41/50).  

Then OR = sqrt(54-4*sqrt(41)) = 5.328

  Posted by Brian Smith on 2016-04-08 11:24:15
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