Final answer:
To find the orthocenter of a triangle, determine the slopes of two sides, calculate the equations of the altitudes, and then solve for their intersection point.
Step-by-step explanation:
To find the orthocenter of a triangle with vertices X(-4, 2), Y(-2, 6), and Z(2, 2), one must calculate the slopes of two sides of the triangle and then determine the equations of the lines perpendicular to these sides that pass through the remaining vertex.
The slope of the line connecting vertices X and Y (let's call this line segment XY) can be calculated using the formula m = (y2 - y1)/(x2 - x1). However, in this case, we want the perpendicular slopes, because altitudes of a triangle, which meet at the orthocenter, are perpendicular to the triangle's sides.
If we take the negative reciprocal of the slope of XY, we get the slope of the altitude from vertex Z. We can do a similar process for another side, for instance, XZ, and find the altitude from vertex Y.
The intersection of these two altitudes will be the orthocenter of the triangle. After finding the equations of these perpendicular lines by plugging in the coordinates of the vertices and slopes, all that remains is to solve these two equations simultaneously to find their point of intersection, which will give us the coordinates of the orthocenter.