Final answer:
The orthocenter of triangle DEF with vertices D(-1, -3), E(-1, 5), and F(5, -1) is the point where the altitudes of the triangle intersect, which is calculated to be at the coordinates (-1, -1).
Step-by-step explanation:
To find the orthocenter of triangle DEF with vertices D(-1, -3), E(-1, 5), and F(5, -1), we need to find the point where the altitudes of the triangle intersect. An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. In this case, since vertices D and E have the same x-coordinate, DE is a vertical line, and thus the altitude from F is the horizontal line through F. Therefore, the altitude from F is the line y = -1.
The altitude from vertex D can be found by determining the slope of EF. Because D is on a vertical line that intersects EF at a right angle, the slope of the altitude from D would be the negative reciprocal of the slope of EF. The slope of EF (where E is (-1, 5) and F is (5, -1)) is:
m = (y2 - y1) / (x2 - x1) = (-1 - 5) / (5 + 1) = -6 / 6 = -1.
Therefore, the slope of the altitude from D is the negative reciprocal of -1, which is 1. Since D has the coordinates (-1, -3), the equation of the altitude from D can be expressed using point-slope form y - y1 = m(x - x1), giving us y + 3 = 1(x + 1), or y = x.
We now have equations for two altitudes: y = x (altitude from vertex D) and y = -1 (altitude from vertex F). To find their intersection point or the orthocenter, we set the equations equal:
y = x
y = -1
Thus, at the intersection, x must also be -1. Therefore, the orthocenter of triangle DEF is at (-1, -1).