Find the coordinates of the circumcenter of triangle DEF with D(5,-1), E(-1,3), F(3,-1).
To find the circumcenter of triangle DEF, we need to find the point that is equidistant from each of the three vertices.
Let's first find the slopes of the perpendicular bisectors of two sides of the triangle: EF and DF.
The midpoint of EF can be found by taking the average of the x-coordinates and the average of the y-coordinates:
Midpoint of EF: ([(3+(-1))/2], [(-1+3)/2]) = (1,1)
The slope of EF is:
mEF = (3-(-1))/(-1-3) = 4/(-4) = -1
The slope of a perpendicular bisector is the negative reciprocal of the slope of the line it is perpendicular to. So, the slope of the perpendicular bisector of EF is 1.
Using the midpoint (1,1) and the slope 1, we can find the equation of the perpendicular bisector of EF:
y - 1 = 1(x - 1)
y - 1 = x - 1
y = x
Now, let's find the midpoint of DF:
Midpoint of DF: ([(3+5)/2], [(-1+(-1))/2]) = (4, -1)
The slope of DF is:
mDF = (-1-(-1))/(5-3) = 0
The slope of a perpendicular bisector is the negative reciprocal of the slope of the line it is perpendicular to. So, the slope of the perpendicular bisector of DF is undefined.
Since the slope is undefined, the equation of the perpendicular bisector of DF is a vertical line passing through the midpoint (4, -1). So, the equation is x = 4.
Now, we have two equations: y = x (from the perpendicular bisector of EF) and x = 4 (from the perpendicular bisector of DF).
To find the circumcenter of DEF, we need to find the point where these two lines intersect. Solving the system of equations, we have:
x = 4
y = 4
So, the circumcenter of triangle DEF is (4, 4).