To find the circumcenter of a triangle, what we want to do is find the midpoint of two of the line segments, find the perpendicular line to the line segments at the midpoint, and see where they meet. The two line segments out of the three in the triangle do not matter.
We have AB, AC, and BC as our three line segments in the triangle (there are no other possibilities!). To find the midpoint of line AB, we have to find the average of the x values and the average of the y values. To find the average of two numbers, simply add them up and divide by 2 (if there were 3 numbers, you’d divide by 3). Finding the midpoint of AB, we get (8+4)/2=6 for the x value (note that the x value comes first in a point!) and (8-8)/2=0 for the y value, making our midpoint (6,0). Using the strategy for BC, we get (4+8)/2=6 and (8+8)/2=8 to get (6,8) as the midpoint. Next, to find the slope of line AB, we subtract the second y value from the first y value and divide that by the value of the second x value subtracted by the first x value, or
Note that if you replace all the 2’s with ones (or vice versa), you still end up with the same answer. Plugging that in for AB, we get
Finding the perpendicular line at that point, we start by finding the negative of the slope , which is -4. Next, we find the reciprocal, or switch the numerator and denominator - in this case, it would be -4/1, which translates into -1/4. Plugging that into the midpoint, we get y=(-1/4)x+b in the equation y=mx+b with m being the slope. If 6 is the x value and y=0, we have 0=-1.5+b. Adding 1.5 to both sides to separate b, we get 1.5=b, making the equation y=(-1/4)x+1.5 For BC, we get the slope to be (8-8)/(4-8)=0. For the negative reciprocal of 0, since 0 does not have a negative nor a reciprocal (0/1 cannot be 1/0 since 0 in the denominator is undefined), we have undefined as a slope. A slope of 0 means a horizontal line and a slope of undefined means a vertical line. Since a horizontal line is of the equation y=b and a vertical line has the equation x=b, we plug (6,8) into it to get 6=b, or x=6 as our line. Lastly, we want to see where x=6 and y=(-1/4)x+1.5 meet. Plugging x=6 into the latter equation, we get -1.5+1.5=0 as our y value, making (6,0) our circumcenter. Feel free to ask further questions!