I really need help with this.

9514 1404 393

  a.  (1.4, 2.6)

  b.  (3, 2)

Step-by-step explanation:

a. The circumcenter is the intersection point of the perpendicular bisectors of the sides of the triangle.

For points X(a, b) and Y(c, d), the perpendicular bisector of XY can be written as ...

  (c -a)(x -(c+a)/2) +(d -b)(y -(d+b)/2) = 0

Then the equation of the perpendicular bisector of MN is ...

  (-2-4)(x -(-2+4)/2) +(4-0)(y -(4+0)/2) = 0

  -6(x -1) +4(y -2) = 0

  3x -2y = -1 . . . .  divide by -2 and put in standard form

Similarly, the perpendicular bisector of NO is ...

  (0-(-2))(x -(0-2)/2) +(6 -4)(y -(6+4)/2) = 0

  2(x +1) +2(y -5) = 0

   x + y = 4 . . . . divide by 2 and put in standard form

So, the point of intersection is ...

  (3x -2y) +2(x +y) = (-1) +2(4) . . . . . add twice the 2nd equation to the 1st

  5x = 7 . . . . . . simplify

  x = 1.4 . . . . . . divide by 5

  y = 4-1.4 = 2.6 . . . . . find y from the second equation

The coordinates of the circumcenter are (1.4, 2.6).

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b. The centroid is the average of the vertex coordinates.

  centroid = (A +B +C)/3 = ((1, 2) +(3, 4) +(5, 0))/3 = (1+3+5, 2+4+0)/3 = (9, 6)/3

  centroid = (3, 2)

because they could drop even lower..