Alright! Let's find the orthocenter of the triangle with the given vertices: (5, -2), (-1, 2), and (1, 4).
First, we need to find the slopes of the lines containing each side of the triangle. We can use the formula: slope = (y2 - y1) / (x2 - x1).
The slope of the line passing through (5, -2) and (-1, 2) is: (2 - (-2)) / (-1 - 5) = 4 / -6 = -2/3.
The slope of the line passing through (-1, 2) and (1, 4) is: (4 - 2) / (1 - (-1)) = 2 / 2 = 1.
Now, we need to find the negative reciprocals of these slopes to get the slopes of the altitudes.
The slope of the altitude corresponding to the line through (5, -2) and (-1, 2) is: -1 / (-2/3) = 3/2.
The slope of the altitude corresponding to the line through (-1, 2) and (1, 4) is: -1 / 1 = -1.
Next, we can use the slope-intercept form of a line (y = mx + b) to find the equations of the altitudes.
For the altitude corresponding to the line through (5, -2) and (-1, 2), we can use the point-slope form: y - y1 = m(x - x1). Let's choose the point (5, -2) as our reference point.
The equation of the altitude is: y - (-2) = (3/2)(x - 5).
Simplifying, we get: y + 2 = (3/2)x - 15/2.
For the altitude corresponding to the line through (-1, 2) and (1, 4), we can use the point-slope form again. Let's choose the point (-1, 2) as our reference point.
The equation of the altitude is: y - 2 = (-1)(x - (-1)).
Simplifying, we get: y - 2 = -x - 1.
Now, we have a system of equations formed by the altitudes:
y + 2 = (3/2)x - 15/2