Triangle GHM is isosceles. G (-2.-3), H (-2,7) and the x-coordinate of M is 4.?
Find the five possible values of the -Y coordinate of M. GM=GH, GH=HM.

Question

anonymous · Answer

Let the y-coordinate of M is y.
Then applying in first case
(i) GM = GH 
 then (GM)^2 = (GH)^2
          ((-2-4)^2)+((-3-y)^2) = ((-2+2)^2)+((-3-7)^2)
           6^2 + (y+3)^2 = 0 + 10^2
           (y+3)^2 = 100-36=64=8^2
            y+3 = 8  or  y+3 = -8
            y = 5 or y= -11
  Second Case
(ii) GH = HM
      (GH)^2 = (HM)^2
      ((-2+2)^2)+((-3-7)^2) = ((-2-4)^2)+((7-y)^2)
       0 + 10^2 = 6^2 + (7-y)^2
            (7-y)^2= 100-36 = 64 = 8^2
             7-y = 8  or  7-y = -8
              y = -1  or y = 15
 Third Case
(iii) GM = HM
       (GM)^2 = (HM)^2
       ((-2-4)^2)+((-3-y)^2) = ((-2-4)^2)+((7-y)^2)
        36 + (y+3)^2 = 36 + (y-7)^2
         y + 3 = y-7 or y + 3 = 7 - y 
          3 = -7 not possible 
         2y = 4
          y = 2.

  Therefore y can have 5 values  -11, -1, 2, 5, 15.