Finding the third side of a triangle given the areaPerimeter = ( a + b + c ) 2 Area = A = p ( p − a ) ( p − b ) ( p − c ) How do I simplify the above two equations to solve for c?

Question

Finding the third side of a triangle given the areaPerimeter =            (      a      +      b      +      c      )        2  Area =   A  =      p    (    p    −    a    )    (    p    −    b    )    (    p    −    c    )  How do I simplify the above two equations to solve for c?

treccinair · Accepted Answer

By cosine rule,                    (1)                              c          2                                    =                  a          2                +                  b          2                −        2        a        b        cos        ⁡        γ        ,            And from the formula for the area                    2        S                            =        a        b        sin        ⁡        γ        ,                            sin        ⁡        γ                            =                              2            S                                a            b                          ,            so, with                    cos        ⁡        γ                            =        ±                  1          −                      sin            2                    ⁡          γ                =        ±                  1          −                                    4                              S                2                                                                    a                2                                            b                2                                                    =        ±                                            a              2                                      b              2                        −            4                          S              2                                            a            b                              (1) becomes                              c          2                                    =                  a          2                +                  b          2                ±        2                              a            2                                b            2                    −          4                      S            2                          .

Holetaug · Accepted Answer

s   is the conventional way of representing the semiperimeter             a      +      b      +      c        2   of the triangle with sides   a    ,  b    ,  c      s    (    s    −    a    )    (    s    −    b    )    (    s    −    c    )    =            −              a        4            −              b        4            −              c        4            +      2              b        2                    c        2            +      2              c        2                    a        2            +      2              a        2                    b        2              4    =  ΔSolving for   c,  c  =            a      2        +          b      2        ±    2                  a        2                    b        2            −      4              Δ        2            where area   =  Δ

Finding the third side of a triangle given the area Perimeter = ( a

Answered question

Answer & Explanation

New Questions in Trigonometry