To find the coordinates of the orthocenter of a triangle, we need to first determine the altitudes of the triangle. The orthocenter is the point where the three altitudes intersect.
Here is how we can find the coordinates of the orthocenter of the triangle with vertices at A(4, -2), B(4, 6), and C(-2, 4):
1. Determine the slopes of the triangle's sides:
- The slope of side AB is (6 - (-2)) / (4 - 4) = 8 / 0 = undefined (vertical line).
- The slope of side BC is (4 - 6) / (-2 - 4) = -2 / -6 = 1/3.
- The slope of side AC is (4 - (-2)) / (-2 - 4) = 6 / -6 = -1.
2. Find the slopes of the altitudes:
- The slope of the altitude from vertex A is the negative reciprocal of the slope of side BC, which is -1/3.
- The slope of the altitude from vertex B is the negative reciprocal of the slope of side AC, which is 1.
- The slope of the altitude from vertex C is the negative reciprocal of the slope of side AB, which is 0 (undefined).
3. Use the slope-intercept form of a line (y = mx + b) to find the equations of the altitudes:
- The equation of the altitude from vertex A passing through A(4, -2) is y = (-1/3)x - (10/3).
- The equation of the altitude from vertex B passing through B(4, 6) is y = x - 2.
- The equation of the altitude from vertex C passing through C(-2, 4) is x = -2.
4. Solve the system of equations formed by the three altitude lines to find the point of intersection (the orthocenter):
- Solving the equations y = (-1/3)x - (10/3), y = x - 2, and x = -2, we find that the orthocenter has the coordinates (2, -2).
Therefore, the coordinates of the orthocenter of the triangle with vertices A(4, -2), B(4, 6), and C(-2, 4) are (2, -2).
