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Cos Case Concern (Posted on 2015-12-06) Difficulty: 3 of 5
O is the center of a circle. AB, CD and EF are three parallel chords of this circle having respective lengths 2, 3 and 4.
It is known that ∠AOB = m, ∠COD = n and, ∠EOF = m+n, where m+n < 180o

Determine cos m

No Solution Yet Submitted by K Sengupta    
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Solution Two solutions | Comment 1 of 4
What is really needed is the radius of the circle since by the law of cosines cos(m)=(2r²-4)/(2r²)

(I first used the law of cosines for cos(n) and cos(n+m) and then the cosine of a sum formula gives and equation.  Solving this by graphing calculator I found r².  But the following is a better way.)

If you rotate the chords the central angles don't change.  So go ahead and rotate them until a triangle of sides 2, 3, 4 is formed.  Circle O is then the circumcircle for this triangle.  Here's a nifty formula for the radius of the circumcircle:  r = abc/(4*area)
By Heron's formula the area = √(135/16)
which yields
r²=64/15
and
cos(m)=17/32

(Which is also what I found doing it the hard way.)

  Posted by Jer on 2015-12-06 12:55:48
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