What is the circumcenter of the triangle with vertices A(0,0) B(0,4) and C(8,0)(1 point)
To find the circumcenter of a triangle, we need to find the intersection of the perpendicular bisectors of the sides of the triangle.
First, let's find the equation of the perpendicular bisector of AB:
The midpoint of AB is ((0+0)/2, (0+4)/2) = (0, 2).
The slope of AB is (4-0)/(0-0) = undefined (since the denominator is 0).
Thus, the slope of the perpendicular bisector of AB is 0.
The equation of the perpendicular bisector passing through (0, 2) is y = 2.
Next, let's find the equation of the perpendicular bisector of BC:
The midpoint of BC is ((0+8)/2, (4+0)/2) = (4, 2).
The slope of BC is (0-4)/(8-0) = -1/2.
Thus, the slope of the perpendicular bisector of BC is 2.
Using the point-slope form, the equation of the perpendicular bisector passing through (4, 2) is y - 2 = 2(x - 4), which simplifies to y = 4x - 6.
Now, let's find the intersection of the two perpendicular bisectors (the circumcenter):
Set y = 2 in the second equation: 2 = 4x - 6.
Solving this equation gives x = 2.
Substituting this value back into the first equation, we get y = 2.
Therefore, the circumcenter of triangle ABC is (2, 2).