Question

5 A triangular park has vertices at A(-1,-1), B(4,4) and C(7,-1). A fountain is to be built that is equidistant from all vertices. Find the location D of the fountain.

Answer 1

Answer:

the coordinates of the triangle are:

A(-1,-1),  B(4,4) and C(7,-1)

Let the D be the point at which the fountain should be planted

then  

the position of the point D will be the centroid of the triangles.

i.e.  x = (x1 + x2 + x3) / 3     and          y = (y1 + y2 + y3) / 3  

      x = (-1 + 4 + 7) / 3          and          y = [-1 + 4 + (-1)] / 3

      x = 10 / 3                       and         y  = [ -1 + 4 -1 ]/3

                                            and         y = (4 - 2) / 2

                                            and         y = 2 / 2

                                            and         y = 1

therefore, the point D(10 / 3 , 1)

Answer 2

Answer:

Step-by-step explanation:

Any point in the perpendicular bisector will be equidistant from two other points, so if we find the intersection of two perpendicular bisectors then the point at the intersection will be equidistant from three points or vertices.

First find the equation of the perpendicular bisector of two of the sides of the triangular park. (It does not matter which two)

AB: Midpoint = ((4-1)/2, (4-1)/2) = (1.5, 1.5)

      Slope = (4+1)/(4+1) = 1

      Perpendicular slope = -1  (Opposite reciprocal)

      y = mx + b    (Substitute midpoint and the perpendicular slope)

      1.5 = -1.5 + b    

      b = 3

      y = -x + 3

AC: Midpoint = ((7-1)/2, (-1-1)/2) = (3, -1)

      Slope = (-1+1)/(7+1) = 0

      Perpendicular slope = undefined (vertical line)

      y = mx + b

      x = 3  (which is a vertical line)

Substitute x for y = -x + 3

y = -3 + 3 = 0

Answer: (3, 0)