Page 176 - Math Course 3 (Book 2)
P. 176

Circles: Equations





         MO. 11 - L5b                                      The center, C, appears to be at (4, 1). This is the
                                                           location of the tower. Find r by using the Distance
            Problem-Solving: Graph the                     Formula with the center and any of the three
                                                           points.
                  Equation of a Circle

                                                               r =  (–1–4)² + (0 – 1)²

                      Let’s Begin                               = 26

                                                            Write an equation.


                                                                                26
        Real World Example                                  (x – 4)² + (y – 1)² = (            )²
                                                            (x – 4)² + (y – 1)² = 26
         ELECTRICITY
         Strategically located substations are extremely
         important in the transmission and distribution of   Check
         a power company’s electric supply. Suppose three   You can verify the location of the center by finding
         substations are modeled by the points D(3, 6),    the equations of the two bisectors and solving a
         E(–1, 0), and F(3, –4). Determine the location of a   system of equations. You can verify the radius by
         town equidistant from all three substations, and   finding the distance between the center and
         write an equation for the circle.                 another of the three points on the circle.

                                                                  c = (4, 1)       midpoint of ED = (1, 3)

                                                                   3 – 1     –2
                                                           slope =        =
                                                                   1 – 4     3
                                                                   –2
                                                            y – 1 =     (x – 4)
                                                                    3
                                                                   –2       11
                                                                   y =   3  x +  3
         Explore You are given three points that lie on a
                  circle.

         Plan     Graph ΔDEF. Construct the perpendicular                          midpoint of DF = (3, 1)
                  bisectors of two sides to locate the cen-         1 – 1
                  ter, which is the location of the tower. Find   slope =                = 0
                                                                    4 – 3
                  the length of a radius. Use the center and
                  radius to write an equation.                     y = 1

                                                                   –2       11
         Solve    Graph ΔDEF and construct the                    1 =   3  x +  3
                  perpendicular bisectors of two sides.
                                                                  3 = –2x + 11

                                                               2x = 8

                                                                 x = 4
                                                                (1, 4)


                                                              r =  (3 – 4)² + (6 – 1)²  = 26





                                                               Answer        (4,1); (x – 4)² +  (y – 1)²  = 26



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