What is the orthocenter of a triangle with corners at #(4 ,5 )#, #(8 ,3 )#, and (7 ,9 )#?

To find the orthocenter of a triangle, you need to determine the point where the altitudes of the triangle intersect. An altitude is a line that passes through one vertex of the triangle and is perpendicular to the opposite side.

1. First, find the slopes of the sides of the triangle. The slope of a line passing through two points ((x_1, y_1)) and ((x_2, y_2)) is given by (m = \frac{y_2 - y_1}{x_2 - x_1}).

   Slope of the line passing through (4, 5) and (8, 3): [m_1 = \frac{3 - 5}{8 - 4} = \frac{-2}{4} = -\frac{1}{2}]

   Slope of the line passing through (8, 3) and (7, 9): [m_2 = \frac{9 - 3}{7 - 8} = \frac{6}{-1} = -6]

   Slope of the line passing through (7, 9) and (4, 5): [m_3 = \frac{5 - 9}{4 - 7} = \frac{-4}{-3} = \frac{4}{3}]

2. The slopes of the altitudes are negative reciprocals of the slopes of the sides that they are perpendicular to. So, the slopes of the altitudes passing through the vertices are (m_1' = \frac{1}{2}), (m_2' = \frac{1}{6}), and (m_3' = -\frac{3}{4}).

3. Use the point-slope form of the equation of a line ((y - y_1) = m(x - x_1)) to find the equations of the altitudes passing through each vertex.

   For the altitude passing through (4, 5): [y - 5 = \frac{1}{2}(x - 4)]

   For the altitude passing through (8, 3): [y - 3 = \frac{1}{6}(x - 8)]

   For the altitude passing through (7, 9): [y - 9 = -\frac{3}{4}(x - 7)]

4. Solve the system of equations formed by the three altitudes to find the point of intersection, which is the orthocenter. The solution is (H(6, 6)).

Therefore, the orthocenter of the triangle with vertices at (4, 5), (8, 3), and (7, 9) is (6, 6).