The center of the circle that circumscribes a triangle is called the circumcenter. The circumcenter is where the perpendicular bisectors of the three sides of the triangle intersect. But you only have to find where two of the perpendicular bisectors intersect - the third will also intersect there.
Pick a side, find the midpoint, then find the slope and equation for its perpendicular bisector. Do that for two sides, then set those equations equal to eachother to solve for x. Substitute that x back into either equation to find the y coordinate.
For this example let's start with side AB. The midpoint has x,y coordinates that are the average of the x and y coordinates of the end points. So add them, and divide by 2.
Midpoint (AB) has x coordinate (5+(-4))/2 = 1/2 and y coordinate (2+(-3))/2 = -1/2. The midpoint is (1/2 , -1/2).
Still working with AB, find the slope of its perpendicular bisector. That slope will be the negative reciprocal of the slope of AB. The slope of AB is (y2 - y1)/(x2-x1) = (-3 - 2)/(-4-5) = -5/-9 = 5/9. The slope of the perp. bisector is -9/5.
The equation of the line is y=mx+b where m is the slope. So it is y = (-9/5)x + b. To find b, substitute in the x and y values for a point we know the line contains. We know it contains the point (1/2 , -1/2).
-1/2 = (-9/5) * 1/2 + b
-1/2 = -9/10 + b
-1/2 + 9/10 = b = -5/10 + 9/10 = 4/10 = 2/5.
The equation of the line is y = (-9/5)x + 2/5
Find the same information for another perp. bisector. This time let's use AC.
MIdpoint x coord. is (5+3)/2 = 8/2 = 4 and midpoint y coord. is 2+-2 / 2 = 0/2 = 0. The midpoint is (4,0).
The slope of AC is (-2-2)/(3-5) = -4/-2 = 2 or 2/1 and thus the slope of the perp. bisector is =1/2.
The equation of the line is y = (-1/2)x + b and we can substitute x,y from the point we know (4,0)
0 = (-1/2)*4 + b
2=b so the equation of that line is y = (-1/2)x + 2.
Now set those two line equations (of the perp. bisectors) equal to each other to find out where they intersect.
(-1/2)x + 2 = (-9/5)x + 2/5
(9/5)x - (1/2)x = 2/5 - 2
(18/10)x - (5/10)x = 2/5 - 10/5
(13/10)x = -8/5
x = -8/5 * 10/13 = -16/13 = -1 and 3/13
Plug that into either equation to find the y coordinate of the circumcenter.
y = (-1/2)* -16/13 + 2 = 16/26 + 2 = 16/26 + 52/26 = 68/26 = 34/13 = 2 and 8/13.
The center of the circumscribed circle, aka the circumcenter of the triangle, is (-16/13, 34/13). You can verify it by graphing the triangle. Since it is an obtuse triangle, the circumcenter should (and does) lie outside the triangle.
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