Let's plot first the given coordinates of the triangle for us to visualize this better.
The circumcenter is the intersection of the three bisectors of the triangle. Let's determine the midpoint and the slope of each bisector so that we can determine the equation of the bisector.
Let's start calculating the midpoint of P and Q. The formula is:
[tex](x_m,y_m)=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]
Our Point P (-2, 5) will be our first point and Point Q (4, 1) will be our second point. Let's plug these x and y values to the formula above.
[tex]\begin{gathered} (x_m,y_m)=(\frac{-2+4}{2},\frac{5+1}{2}) \\ (x_m,y_m)=(\frac{2}{2},\frac{6}{2}) \\ (x_m,y_m)=(1,3) \end{gathered}[/tex]
Hence, the midpoint of PQ is (1, 3). Now, let's determine the equation of the bisector from R to the PQ's midpoint. Let's use the two-point formula.
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]
Our first point will be the midpoint of PQ (1, 3) and the second point will be point R at (-2, -3). Let's plug these x and y values to the formula above.
[tex]y-3=\frac{-3-3}{-2-1}(x-1)[/tex]
And solve for y.
[tex]\begin{gathered} y-3=\frac{-6}{-3}(x-1) \\ y-3=2(x-1) \\ y-3=2x-2 \\ y=2x-2+3 \\ y=2x+1 \end{gathered}[/tex]
Hence, the equation of the line bisecting the segment PQ from Point R is y = 2x + 1.
Now, let's move on to the midpoint of QR. First Point Q (4, 1) and second Point R (-2, -3).
[tex]\begin{gathered} (x_m,y_m)=\frac{4+(-2)}{2},\frac{1+(-3)}{2} \\ (x_m,y_m)=\frac{2}{2},-\frac{2}{2} \\ (x_m,y_m)=1,-1 \end{gathered}[/tex]
The midpoint of QR is at (1, -1). Let's determine the equation of the bisector from Point P (-2, 5) to the midpoint of QR.
[tex]\begin{gathered} y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1) \\ y-(-1)=\frac{5-(-1)}{-2-1}(x-1) \\ y+1=\frac{6}{-3}(x-1) \\ y+1=-2(x-1) \\ y+1=-2x+2 \\ y=-2x+2-1 \\ y=-2x+1 \end{gathered}[/tex]
The equation of the bisector from Point P to midpoint of QR is y = -2x + 1.
Let us show the graph of these bisectors in the triangle.
