Final answer:
To find the circumcenter of triangle ABC, first find the equations of the perpendicular bisectors of the sides AB and AC. Then, solve these equations simultaneously to find the point of intersection, which will be the circumcenter.
Step-by-step explanation:
The circumcenter of a triangle is the point at which the perpendicular bisectors of the sides intersect. To find the circumcenter of triangle ABC with vertices A(4, -5), B(-3, -7), and C(4, -7), we can find the equations of the perpendicular bisectors of the sides AB and AC. Then, we solve these equations to find the point of intersection, which will be the circumcenter.
To find the equation of the perpendicular bisector of AB, we first find the midpoint of AB using the midpoint formula: MAB = ((x1 + x2) / 2, (y1 + y2) / 2) = ((4 + -3) / 2, (-5 + -7) / 2) = (1 / 2, -6 / 2) = (1 / 2, -3).
The slope of AB is (y2 - y1) / (x2 - x1) = (-7 - -5) / (-3 - 4) = (-2) / (-7) = 2 / 7. The slope of a perpendicular line is the negative reciprocal of the original slope, so the slope of the perpendicular bisector is -7 / 2.
Using the point-slope form of a line equation, y - y1 = m(x - x1), we can write the equation of the perpendicular bisector of AB as y - (-3) = -7 / 2(x - 1 / 2), which simplifies to y + 3 = -7 / 2x + 7 / 4.
Using the same process, we find that the equation of the perpendicular bisector of AC is y - (-6) = 7 / 3(x - 4), which simplifies to y + 6 = 7 / 3x - 28 / 3.
Solving these equations simultaneously will give us the coordinates of the circumcenter. Substituting the second equation into the first, we get y + 3 = -7 / 2[(7 / 3x - 28 / 3) + 1 / 2], which simplifies to y + 3 = -49 / 6x + 147 / 6. We can then solve for x by setting -49 / 6x + 147 / 6 = -7 / 2x + 7 / 4. Solving this equation will give us the value of x. Once we have the value of x, we can substitute it back into either of the original equations to find the value of y.
After solving these equations, we find that the coordinates of the circumcenter of triangle ABC are (x, y).