Final answer:
The orthocenter of a triangle with vertices (4, -3), (8, 5), and (8, -8) is the point (8, -1.5) where the altitude from A to BC intersects.
Step-by-step explanation:
The orthocenter of a triangle is the point of intersection of the altitudes of the triangle. To find the orthocenter, we need to find the equations of the altitudes.
Let's label the given points as A(4, -3), B(8, 5), and C(8, -8).
The equation of the midpoint of BC can be found using the midpoint formula: M_bc = ((x_b + x_c)/2, (y_b + y_c)/2) = ((8 + 8)/2, (5 + -8)/2) = (8, -1.5).
The equation of the line BC can be found using the slope-intercept form: y - y_b = m_bc(x - x_b), where m_bc = (y_c - y_b)/(x_c - x_b).
The slope of BC is m_bc = (-8 - 5)/(8 - 8) = -13/0. The line BC is vertical, so the equation becomes x = 8.
The equation of the altitude from A to BC can be found using the point-slope form: y - y_a = m_ab(x - x_a), where m_ab = -1/m_bc.
Using the coordinates of point A and the slope m_ab, the equation of the altitude from A to BC is y - (-3) = -1/(-13/0)(x - 4), which simplifies to y + 3 = 0(x - 4), or y + 3 = 0.
Therefore, the orthocenter of the triangle with vertices A(4, -3), B(8, 5), and C(8, -8) is the point (8, -1.5) where the altitude from A to BC intersects.