The area of a triangle with sides 59 , 37 , 12 5

Question

The area of a triangle with sides   59  ,  37  ,  12      5

conveneau71 · Accepted Answer

Step 1One hour ago i just answered a similar problem, an equivalent version of Heron&#039;s formula applies, and the computation is just one line. The formula for the area A of the given triangle can be extracted from:  16      A    2    =  (      a    2    +      b    2    +      c    2        )    2    −  2  (      a    4    +      b    4    +      c    4    )     , ,in our case we have       a    2    =  3481 ,       b    2    =  1369 ,       c    2    =  720 , so  16      A    2    =  (  3481  +  1369  +  720      )    2    −  2  (      3481    2    +      1369    2    +      720    2    )  =  2005056  =  16  ⋅  (  2  ⋅  3  ⋅  59      )    2       .So   A  =  2  ⋅  3  ⋅  59  =      354   .Step 2Well, now after knowing the answer, and seeing the factor 59 in A, it becomes natural to compute the height corresponding to the (biggest) side with length 59. Let h be this (smallest) height, and let x, y be the lengths of the projections of the two other sides, x corresponding to 37, y to   12      5   , on this biggest side. Then we can immediately write the system joining x, y, h:      {                            59                                    =          x          +          y                                                  37            2                                              =                      x            2                    +                      h            2                                                5          ⋅                      12            2                                              =                      y            2                    +                      h            2                                  Subtracting the last two equation from each other we get  11  ⋅  59  =  649  =  1369  −  720  =      x    2    −      y    2    =  (  x  +  y  )  (  x  −  y  )     ,so   x  −  y  =  11 , then   x  =  (  59  +  11  )      /    2  =  35 ,   y  =  (  59  −  11  )      /    2  =  24 if needed for checks,       h    2    =      37    2    −      35    2    =  (  37  +  35  )  (  37  −  35  )  =  72  ⋅  2  =  144  =      12    2   , so   h  =  12 , leading to the known answer for the area,             1      2        ⋅    12    ⋅    59    =          354       .

Logan Wyatt · Accepted Answer

Step 1Draw altitude CH. Then:  H  A  =  59  −      C    1    H  144  ×  5  −      C    1        H    2    =      37    2    −  H      A    2    =  C      H    2  From here find       C    1    H . Then you can find CH and so the area.

The area of a triangle with sides 59

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