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 Equilateral fun (Posted on 2015-02-28)
In an equilateral triangle ABC on the side AC taken points P and Q such that AP=3, QC=5. (A-->P-->Q-->C). It is known that angle PBQ=30.

Find the side of the triangle ABC

 No Solution Yet Submitted by Danish Ahmed Khan No Rating

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 Solution Comment 2 of 2 |
`Let M be the midpoint of AC andx the side length of the triangle.`
`  1/sqrt(3) = tan(30) = tan(PBQ)`
`            = tan(PBM + MBQ)`
`                tan(PBM) + tan(MBQ)            = -----------------------               1 - tan(PBM)*tan(MBQ)                   x/2-3           x/2-5               ------------- + -------------                x*sqrt(3)/2     x*sqrt(3)/2            = -----------------------------------                       x/2-3           x/2-5                 1 - ------------- * -------------                    x*sqrt(3)/2     x*sqrt(3)/2`
`            or`
`          0 = (x-1)*(x-15)`
`Since x is clearly greater than 1, x = 15.QED`

 Posted by Bractals on 2015-03-01 11:17:33

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