Page 134 - Math Course 3 (Book 2)
P. 134

Proportional Parts of Triangles





          Triangle ABC has vertices A(–2, 2), B(2, 4) and    First, use the Distance Formula to find BC and DE.
          C(4, –4). DE is a midsegment of △ABC.
          Verify that BC || DE.                              BC =  (2 – 4)² + [4 –( –4)]²

                                      B                         = 4 + 64
                             y
                         D               (2, 4)
                                                                = 68    or  2 17

           (–2, 2)
                A
                                                             DE =  (0 – 1)² + [3 –( –1)]²

                                                                = 1 + 16
                         O                       x

                             E                                  = 17


                                                                        DE        √17       1
                                              (4, –4)                   BC   =  2 √17   or  2
                                             C


                                                                                    1
                                                                                                 1
                                                                             DE
         If the slopes of BC and DE are equal, BC || DE.      Answer       If          =       , then DE =        BC.
                                                                                    2
                                                                                                 2
                                                                             BC
                      – 4 – 4
         slope of BC =                 or – 4
                       4 – 2
                      –1 – 3
         slope of DE =                 or – 4
                       1 – 0
                                                           Congruent Segments
                         Because the slopes BC and DE
            Answer
                                 of BC || DE.
                                                              Example
                                                           Find x and y.
          Triangle ABC has vertices A(–2, 2), B(2, 4) and
          C(4, –4). DE is a midsegment of △ABC.
                        1
          Verify that DE =      BC.
                        2                                                  2x + 2        3x – 4
                                     B
                             y
                         D              (2, 4)


           (–2, 2)
                A                                                 8  y + 7
                                                                  3
                                                                               5y
                        O                        x
                             E
                                                           To find x:


                                              (4, –4)        2x + 2 = 3x – 4  Given
                                             C                      2 = x – 4  Subtract 2x from each side.

                                                                 6 = x      Add 4 to each side.



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