find the coordinates of the circumcenter of tringleABC

A(3,-1),B(-2,-1),C(3,-8)

the circumcentre lies on the intersection of the right-bisectors of any two chords.

I noticed that AB forms a horizontal line, so that one will be easy.
midpoint is (1/2 , -1) , so the equation of the right-bisector will be x = 1/2
I also noticed that AC is a vertical line, so the equation of the right-bisector of AC is y = -9/2

circumcentre is (1/2 , -9/2)

That was simple since Triangle ABC was right-angled with vertical and horizontal sides
Had I seen that right away, I could have simply taken the midpoint of the hypotenuse, which would have been
( ((-2+3)/2 , (-1-8)/2 )
= (1/2 , -9/2)

To find the coordinates of the circumcenter of a triangle, you need to use the properties of perpendicular bisectors. The circumcenter is the point where the perpendicular bisectors of the sides of the triangle intersect.

Step 1: Find the midpoint of each side of the triangle.
To find the midpoint of any two points (x1, y1) and (x2, y2), you can use the midpoint formula:
midpoint(x, y) = ((x1 + x2)/2, (y1 + y2)/2)

Let's find the midpoints of the sides AB, BC, and AC.

Midpoint of AB:
x = (3 + (-2))/2 = 1/2
y = (-1 + (-1))/2 = -1
So, the midpoint of AB is (1/2, -1).

Midpoint of BC:
x = (-2 + 3)/2 = 1/2
y = (-1 + (-8))/2 = -9/2
So, the midpoint of BC is (1/2, -9/2).

Midpoint of AC:
x = (3 + 3)/2 = 6/2 = 3
y = (-1 + (-8))/2 = -9/2
So, the midpoint of AC is (3, -9/2).

Step 2: Find the slopes of the lines passing through each side of the triangle.
The slope of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the slope formula:
slope = (y2 - y1)/(x2 - x1)

Let's find the slopes of the lines passing through AB, BC, and AC.

Slope of AB:
slope = (-1 - (-1))/(-2 - 3) = 0/(-5) = 0

Slope of BC:
slope = (-1 - (-8))/(3 - (-2)) = 7/5

Slope of AC:
slope = (-1 - (-8))/(3 - 3) = 7/0 (Note: The slope is undefined because the difference in x-coordinates is 0. This means the line is vertical.)

Step 3: Calculate the perpendicular slopes.
The perpendicular slope to any given slope is the negative reciprocal of that slope. To find the perpendicular slope, you can take the negative reciprocal of the given slope.

Perpendicular slope to AB: The slope of a line perpendicular to a line with a slope of 0 is undefined.

Perpendicular slope to BC:
perpendicular slope = -1/(7/5) = -5/7

Perpendicular slope to AC: A line perpendicular to a vertical line has a slope of 0.

Step 4: Calculate the equations of the perpendicular bisectors.
Now that we have the midpoints and perpendicular slopes, we can calculate the equations of the perpendicular bisectors.

Equation of the perpendicular bisector of AB:
Using the point-slope form (y - y1) = m(x - x1), where (x1, y1) is the midpoint of AB and m is the perpendicular slope:
(y - (-1)) = (-5/7)(x - (1/2))
This simplifies to:
y + 1 = (-5/7)(x - 1/2)

Equation of the perpendicular bisector of BC:
Using the point-slope form:
(y - (-9/2)) = (0)(x - 1/2)
This simplifies to:
y + 9/2 = 0

Equation of the perpendicular bisector of AC:
Using the point-slope form:
(y - (-9/2)) = (0)(x - 3)
This simplifies to:
y + 9/2 = 0

Step 5: Find the intersection point of the perpendicular bisectors.
Solve the system of equations formed by the equations of the perpendicular bisectors to find their intersection point. In this case, the intersection point is the circumcenter of the triangle.

From the equations of the perpendicular bisectors:
y + 1 = (-5/7)(x - 1/2)
y + 9/2 = 0

Solving these equations, we find that the x-coordinate is 1/2.

Substituting this x-coordinate into any of the equation of the perpendicular bisectors, we find that the y-coordinate is -9/2.

Therefore, the coordinates of the circumcenter of triangle ABC are (1/2, -9/2).