The sides a,b,c of a △ A B C are in GP whose common ratio is 2 3 and the circumradius of the triangle is 6 7 209 .Find the longest side of the triangle.I used law of sines a sin ⁡ A = b sin ⁡ B = c sin ⁡ C = 2 RI took a = a , b = 2 3 a , c = ( 2 3 ) 2 awhich gives sin ⁡ A sin ⁡ B = sin ⁡ B sin ⁡ C I am stuck now,how to find the longest side a

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The sides a,b,c of a   △  A  B  C are in GP whose common ratio is       2    3   and the circumradius of the triangle is   6            7      209      .Find the longest side of the triangle.I used law of sines      a          sin      ⁡      A        =      b          sin      ⁡      B        =      c          sin      ⁡      C        =  2  RI took   a  =  a  ,  b  =      2    3    a  ,  c  =            (              2        3            )        2    awhich gives            sin      ⁡      A              sin      ⁡      B        =            sin      ⁡      B              sin      ⁡      C      I am stuck now,how to find the longest side a

Eliza Flores · Accepted Answer

If the side lengths are   l  ,      2    3    l  ,      4    9    l then the circumradius is given by:  R  =            a      b      c              4      Δ        =                    8        27                    l        3                            l        2                              (          1          +                      2            3                    +                      4            9                    )                          (          1          −                      2            3                    +                      4            9                    )                          (          1          +                      2            3                    −                      4            9                    )                          (          −          1          +                      2            3                    +                      4            9                    )                    by Heron&#039;s formula, hence:  R  =      24          1463        lgives:  l  =            7      4        .

Aliana Kaufman · Accepted Answer

Let a,b,c be the side lengths. Then, b,c can be written as  b  =      2    3    a  ,    c  =            (              2        3            )        2    awhere a is the longest side. So, we have, by the law of cosines,      a    2    =            (              2        3            a      )        2    +            (                        (                      2            3                    )                2            a      )        2    −  2  ⋅      2    3    a  ⋅            (              2        3            )        2    a  cos  ⁡  A  ⇒  cos  ⁡  A  =  −      29    48    .Hence,  a  =  2  R  sin  ⁡  A  =  2  R      1    −          cos      2        ⁡    A    =  2  ⋅  6            7      209            1    −                  (        −                  29          48                )            2        =      7    4    .

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