Final answer:
The circumcenter of Triangle ABC can be found by finding the intersection point of the perpendicular bisectors of the triangle's sides. By finding the midpoints of the sides and the slopes of the perpendicular bisectors, we can determine the equations of the perpendicular bisectors. Solving these equations will give us the coordinates of the circumcenter, which in this case is (-5, 4.5).
Step-by-step explanation:
To find the circumcenter of a triangle, we need to find the intersection point of the perpendicular bisectors of the triangle's sides.
First, we find the midpoints of the sides AB, BC, and AC:
M₁ = ((-7 + (-3))/2 , (7 + 4)/2) = (-5, 5.5)
M₂ = ((-3 + (-7))/2 , (4 + 4)/2) = (-5, 4)
M₃ = ((-7 + (-7))/2 , (7 + 4)/2) = (-7, 5.5)
Next, we find the slopes of the sides AB, BC, and AC:
Slope of AB = (4 - 7)/(-3 - (-7)) = 3/4
Slope of BC = (4 - 4)/(-7 - (-3)) = 0
Slope of AC = (7 - 4)/(-7 - (-7)) = 3/14
Since the slopes of the perpendicular bisectors are negative reciprocals of the slopes of the sides, the slopes of the perpendicular bisectors are -4/3, undefined, and -14/3, respectively.
Finally, we find the equations of the perpendicular bisectors passing through the midpoints:
Equation of perpendicular bisector of AB passing through M₁: y - 5.5 = (-4/3)(x + 5)
Equation of perpendicular bisector of BC passing through M₂: x + 5 = 0
Equation of perpendicular bisector of AC passing through M₃: y - 5.5 = (-14/3)(x + 7)
To find the circumcenter, we solve the system of equations formed by these three equations. The circumcenter of the triangle ABC is (-5, 4.5), which corresponds to option B.