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Equilateral fun (Posted on 2015-02-28) Difficulty: 2 of 5
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    
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Solution Solution Comment 2 of 2 |

Let M be the midpoint of AC and
x 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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