Final answer:
The circumcenter of the triangle with given vertices is (1, 3). The centroid of the triangle with given vertices is (-3.333, 2).
Step-by-step explanation:
The circumcenter of a triangle is the center of a circle that passes through all three vertices of the triangle. To find the circumcenter, we can use the perpendicular bisectors of two sides of the triangle. Let's find the circumcenter of the triangle with vertices A(0,1), B(2, 1), and C(2, 5):
- Find the slopes of the line segments AB and AC.
- Find the midpoints of AB and AC.
- Find the equations of the perpendicular bisectors of AB and AC.
- Find the intersection point of the two perpendicular bisectors, which will be the circumcenter.
After performing these steps, we find that the circumcenter of the triangle is (1, 3).
The centroid of a triangle is the point of intersection of its medians. To find the centroid, we can find the midpoints of the three sides of the triangle. Let's find the centroid of the triangle with vertices A(−6, 0), B(−4, 4), and C(0, 2):
- Find the midpoints of AB, BC, and AC.
- Find the coordinates of the centroid by finding the average of the x-coordinates and the average of the y-coordinates of the midpoints.
After performing these steps, we find that the centroid of the triangle is (−3.333, 2).