Page 138 - Math Course 3 (Book 2)
P. 138

Proportional Relationships of Similar Triangles
          Mo. 10


           Lesson 5                                                                       Proportional
                                                            XY   =    perimeter of△XYZ    Perimeter
                                                            AB        perimeter of △ABC   Theorem

          KEY CONCEPTS:                                     24   =            x           Substitution.
                                                            16            48 + 16√5
          1. Recognize and use proportional
              relationships of corresponding                24 (48 + 16√5) = 16x          Cross products.
              perimeters of similar triangles.
          2. Recognize and use proportional                   1152 + 384√5 = 16x          Multiply.
              relationships of corresponding angle                                        Divide each
              bisectors, altitudes, and medians of                   72 + 24√5 = x        side by 16.
              similar triangles.
                                                                                 The perimeter of
                                                              Answer
         MO. 10 - L5a                                                         △XYZ is 72 + 24√5 units.

            Proportional Perimeters of

                   Similar Triangles                       Write a Proof



         THEOREM 10.7                                         Example


         Proportional Perimeters Theorem                    Write a paragraph proof.
         If two triangles are similar, then the perimeters are
         proportional to the measures of corresponding      Given:  △JKL ~ △QRS
         sides.                                                    MK is a median of △JKL.
                                                                   TR is a median of △QRS.
                                                            Prove:  △JKM ~ △QRT
                      Let’s Begin                                               J





        Perimeters of Similar Triangles
                                                            E                   I                      K
            Example                                                                        L

                                                            D               F
         If △ABC ~ △XYZ, AC = 32, AB = 16, BC = 16√5,               G
         and XY = 24, find the perimeter of △XYZ.

                                                                              Answer
                             B         16√5                 Proof:
                                                                                         JK    JL
                                                                                             =
          Y                                                 By definition of similar triangles                     .
                                                                                         QR
                                                                                               QS
                             16
                                                            Since similar triangles corresponding medians
                                                                                                JL    KM
                                                            proportional to the corresponding sides
                                                                                                   =
                             A        32         C                                              QS    RT
                                                                          JK
         24                                                 By substitution                     Since MK and TR are
                                                                                LM
                                                                              =
                                                                          QR    RT
                                                            medians of △JKL and △QRS, respectively, M and T
          X                  Z
                                                            are midpoints of JL and QS. By definition of
         Let x represent the perimeter of △XYZ. The perimeter   midpoint, 2JM=JL and 2QT = QS.
         of △ABC = 16 + 16                    or 48 + 16  5
                            5 + 32
    130
   133   134   135   136   137   138   139   140   141   142   143