Point A has coordinates ( x 1 , y 1 ), point B has coordinates ( x 2 , y 2 ) andpoint M has coordinates ( x 1 + x 1 2 , y 1 + y 1 2 )Prove that A M = 1 2 A B

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Point   A has coordinates   (            x              1              ,          y              1              ), point   B has coordinates   (            x              2              ,          y              2              ) andpoint   M has coordinates   (                    x                  1                    +              x                  1                      2    ,                    y                  1                    +              y                  1                      2    )Prove that   A  M  =      1    2    A  B

decoratesuw · Accepted Answer

The points   A  (      x    1    ,      y    1    )  ,  B  (      x    2    ,      y    2    )  ,  M      (          x      m        =                            x          1                +                  x          2                    2        ,          y      m        =                            y          1                +                  y          2                    2        )   are given.First note that the points   A,   M and   B are colinear (i.e. lie on the same line):  slope of AM  =                    y        m            −              y        1                            x        m            −              x        1              =                                          y            1                    +                      y            2                          2            −              y        1                                                  x            1                    +                      x            2                          2            −              x        1              =                    y        2            −              y        1                            x        2            −              x        1              =  slope of AB  .The triangles   Δ  A  M  D and   Δ  A  B  C are similar, because their corresponding angles are equal. Hence:            A      M              A      B        =            A      D              A      C        =            A      D              2      A      D        =      1    2    ⇒  A  M  =      1    2    A  B  .

Taniya Burns · Accepted Answer

A                  M          2                                    =                              (                                                            x                  1                                +                                  x                  2                                            2                        −                          x              1                        )                    2                +                              (                                                            y                  1                                +                                  y                  2                                            2                        −                          y              1                        )                    2                                                  =                              (                                          −                                  x                  1                                +                                  x                  2                                            2                        )                    2                +                              (                                          −                                  y                  1                                +                                  y                  2                                            2                        )                    2                                                  =                  1          4                                      (                          (        −                  x          1                +                  x          2                          )          2                +        (        −                  y          1                +                  y          2                          )          2                                      )                          =                  1          4                A                  B          2                ,            from which it follows that   A  M  =      1    2    A  B.

Point A has coorditates (x_1,y_1), point B has coordinates (x_2,y_2) and point M has coordinates((x_(1)+x_(2))/2, (y_(1)+y_(2))/2). Prove that AM=1/2AB.

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