Answer:
(-1,2)
Explanation:
The orthocenter of a triangle is the point where the altitudes of the triangle intersect. An altitude is a line segment drawn from a vertex of the triangle perpendicular to the opposite side.
To find the orthocenter of triangle ABC, we first need to find the equations of the three altitudes.
The altitude from vertex A passes through the midpoint of BC, which is (1-1)/2 = 0, (0+3)/2 = 3/2. The slope of line BC is (3-0)/(-1-1) = -3/2, so the slope of the altitude from A is the negative reciprocal, 2/3. Therefore, the equation of the altitude from A is y - 0 = (2/3)(x + 4), or y = (2/3)x + 8/3.
The altitude from vertex B passes through the midpoint of AC, which is (-4-1)/2 = -5/2, (0+3)/2 = 3/2. The slope of line AC is (3-0)/(-1+4) = 1, so the slope of the altitude from B is the negative reciprocal, -1. Therefore, the equation of the altitude from B is y - 0 = (-1)(x - 1), or y = -x + 1.
The altitude from vertex C passes through the midpoint of AB, which is (-4+1)/2 = -3/2, (0+0)/2 = 0. The slope of line AB is (0-0)/(-4+1) = 0, so the slope of the altitude from C is undefined (since it's perpendicular to the x-axis). Therefore, the equation of the altitude from C is x = -1.
To find the orthocenter, we need to find the point where the three altitudes intersect. One way to do this is to solve the system of equations:
y = (2/3)x + 8/3
y = -x + 1
x = -1
Substituting x = -1 into the second equation gives y = 2. Substituting x = -1 and y = 2 into the first equation gives 2 = (2/3)(-1) + 8/3, which is true. Therefore, the point of intersection is (-1, 2), which is the orthocenter of triangle ABC.
Therefore, the coordinates of the orthocenter of triangle ABC are (-1, 2).