9514 1404 393
Answer:
(0.37, 2.52)
Explanation:
The circumcenter is the point of intersection of the perpendicular bisectors of the sides. For the purpose of writing equations for those, we need to know the difference in coordinates of the sides, and the midpoint.
(∆x, ∆y) = B-A = (4, 5) -(-4, 2) = (4+4, 5-2) = (8, 3)
(mx, my) = (B+A)/2 = ((4, 5) +(-4, 2))/2 = (0, 7/2)
Then the general form equation of the perpendicular bisector can be written as ...
∆x(x -mx) +∆y(y -my) = 0
8(x -0) +3(y -7/2) = 0
8x +3y -21/2 = 0 . . . . collect terms
16x +6y -21 = 0 . . . . . eliminate the fraction
Similarly, the equation for the perpendicular bisector of BC will be ...
(∆x, ∆y) = C -B = (3, -1) -(4, 5) = (-1, -6)
(mx, my) = (C +B)/2 = ((3, -1) +(4, 5))/2 = (7/2, 2)
-1(x -7/2) -6(y -2) = 0
Multiplying by -2, we have ...
2x +12y -31 = 0
__
Solving these equations by the "cross multiplication method", we find ...
x = (6(-31) -(12(-21))/(16(12)-2(6)) = 66/180 ≈ 0.366667
y = (-21(2) -(-31)(16))/180 = 454/180 ≈ 2.52222
Rounded to hundredths, the circumcenter is (0.37, 2.52).