Answer:
To find the circumcenter of the triangle with vertices P(1,3), Q(5,5), and R(7,5), we can find the point of intersection of the perpendicular bisectors of two sides of the triangle.
The midpoint of PQ is (3, 4), and the equation of the perpendicular bisector of PQ is y = -2x + 10. The equation of the perpendicular bisector of QR is x = 6. The intersection of these lines is the circumcenter of the triangle, which is (6, -2).
Therefore, the coordinates of the circumcenter are (6, -2).
Explanation:
To find the circumcenter of a triangle, we need to find the point where the perpendicular bisectors of the sides of the triangle intersect. The circumcenter is equidistant from the three vertices of the triangle.
We can start by finding the equations of the perpendicular bisectors of two sides of the triangle, say, PQ and QR. The midpoint of PQ is ((1+5)/2, (3+5)/2) = (3, 4), and the slope of PQ is (5-3)/(5-1) = 2/4 = 1/2. Therefore, the slope of the perpendicular bisector of PQ is -2 (the negative reciprocal of 1/2), and its y-intercept can be found by plugging in the coordinates of the midpoint: y = -2x + 10.
Similarly, the midpoint of QR is ((5+7)/2, (5+5)/2) = (6, 5), and the slope of QR is (5-5)/(7-5) = 0. Therefore, the slope of the perpendicular bisector of QR is undefined (a vertical line), and its equation is simply x = 6.
Now we can find the point of intersection of the two perpendicular bisectors. Since the second perpendicular bisector is a vertical line with x = 6, we only need to substitute x = 6 into the equation of the first perpendicular bisector to get its y-coordinate:
y = -2(6) + 10 = -2
So the circumcenter of triangle PQR is the point (6, -2).
Therefore, the coordinates of the circumcenter are (6, -2).