Final answer:
To find the coordinates of the orthocenter, we need to find the equation of each altitude of the triangle. The equation of the line passing through F and G is y = x + 4, the equation of the line passing through G and H is y = -1/5x + 26/5, and the equation of the line passing through H and F is y = 1/3x + 2. Solving the system of equations gives the coordinates of the orthocenter as (3,2).
Step-by-step explanation:
To find the coordinates of the orthocenter, we need to find the altitude of each side of the triangle and determine where they intersect. The altitude of a side of a triangle is a line segment that is perpendicular to that side and passes through the opposite vertex. Let's find the equation of each altitude:
First, find the slope of the line passing through F (-3,1) and G (1,5). The slope is (5-1)/(1-(-3)) = 4/4 = 1. The equation of the line is y - 1 = 1(x - (-3)) => y - 1 = x + 3 => y = x + 4.
Next, find the slope of the line passing through G (1,5) and H (6,4). The slope is (4-5)/(6-1) = -1/5. The equation of the line is y - 5 = -1/5(x - 1) => y - 5 = -1/5x + 1/5 => y = -1/5x + 26/5.
Lastly, find the slope of the line passing through H (6,4) and F (-3,1). The slope is (1-4)/(-3-6) = -3/-9 = 1/3. The equation of the line is y - 4 = 1/3(x - 6) => y - 4 = 1/3x - 2 => y = 1/3x + 2.
To find the coordinates of the orthocenter, solve the system of equations formed by the three altitude lines. The solution is (3,2), so the coordinates of the orthocenter are (3,2).