Answer:
To find the circumcenter of a triangle, we need to find the point where the perpendicular bisectors of the sides of the triangle intersect.
Let's start by finding the equation of the line passing through the midpoint of the segment AB and perpendicular to AB. The midpoint of AB is ((2+8)/2, (4+2)/2) = (5,3), and the slope of AB is (2-4)/(8-2) = -1/3. The slope of a line perpendicular to AB is the negative reciprocal of -3, which is 3. So, the equation of the line passing through (5,3) and perpendicular to AB is y - 3 = 3(x - 5), or y = 3x - 12.
Similarly, we can find the equation of the line passing through the midpoint of BC and perpendicular to BC. The midpoint of BC is ((8+4)/2, (2-2)/2) = (6,0), and the slope of BC is (-2-2)/(4-8) = 1/2. The slope of a line perpendicular to BC is the negative reciprocal of 1/2, which is -2. So, the equation of the line passing through (6,0) and perpendicular to BC is y - 0 = -2(x - 6), or y = -2x + 12.
Finally, we can find the equation of the line passing through the midpoint of AC and perpendicular to AC. The midpoint of AC is ((2+4)/2, (4-2)/2) = (3,1), and the slope of AC is (-2-4)/(4-2) = -3/2. The slope of a line perpendicular to AC is the negative reciprocal of -3/2, which is 2/3. So, the equation of the line passing through (3,1) and perpendicular to AC is y - 1 = (2/3)(x - 3), or y = (2/3)x - 1/3.
Now, we need to find the intersection point of these three lines, which will give us the circumcenter of the triangle. To do this, we can solve the system of equations:
y = 3x - 12
y = -2x + 12
y = (2/3)x - 1/3
Substituting the first equation into the second equation, we get:
3x - 12 = -2x + 12
5x = 24
x = 24/5
Substituting x = 24/5 into the first equation, we get:
y = 3(24/5) - 12 = 36/5
So, the circumcenter of the triangle is the point (24/5, 36/5)