The orthocenter of a triangle is the point where its three altitudes meet. If we draw a right triangle, the orthocenter will be the vertex of the right triangle, since two of the altitudes are the legs of the right triangle:
If the triangle is acute, the orthocenter lies inside the triangle. If the triangle is obtuse, the orthocenter will lie outside of it.
The orthocenter can be identified by finding the intersection of any two altitudes since the third one will also intersect the others at that point:
For this triangle, we found the orthocenter as follows:
The altitude relative to the side AB of the triangle is the horizontal line passing through the vertex with y-coordinate 2. So, it's the line y = 2.
The altitude relative to the side AC is the line perpendicular to AB and passing through the vertex (0, 4).
Since the line trough AB has slope 2/4 = 1/2, its perpendicular lines have slope -2. So, the line trough altitude relative to AB is:
y = -2x + 4
So, the orthocenter, where those two altitudes intersect, is the solution to the system of equations:
y = 2
y = -2x + 4
Using the first equation into the second, we find:
2 = -2x + 4
2 + 2x = -2x + 4 + 2x
2 + 2x = 4
2 + 2x - 2 = 4 - 2
2x = 2
x = 1
Therefore, the orthocenter of this triangle is the point (1, 2).
