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 Cos Case Concern (Posted on 2015-12-06)
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 No Rating

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 Solution | Comment 3 of 4 |
`Let the radius of the circle be R.`
`Applying the law of cosines to triangle AOBwe get          cos(m) = 1 - 2^2/(2*R^2)         (1)`
`So all we need is R^2.`
`Let G be the point on the smaller arc EFsuch that |GF| = |AB| = 2.`
`   |GF| = 2  ==>  /GOF = /AOB = m             ==>  /EOG = /EOF - /GOF             ==>  /EOG = (m+n) - m = n             ==>  |EG| = 3`
`Therefore, R is the circumradius of triangleEFG whose side lengths are 2, 3, and 4.`
`The circumradius squared of a triangle whoseside lengths are a, b, and c is`
`                (a*b*c)^2   R^2 = ------------------------          16*s*(s-a)*(s-b)*(s-c)`
`,where s is its semiperimeter.`
`Plugging in 2, 3, and 4 for a, b, and c we get`
`   R^2 = 64/15`
`Plugging this into equation (1) gives`
`   cos(m) = 17/32 = 0.53125`
`QED`

 Posted by Bractals on 2015-12-06 15:46:04
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