Final answer:
To find the orthocenter coordinates of a triangle, determine the altitudes' equations by using the negative reciprocals of the slopes of the sides opposite the vertices. Then, find the intersection of the altitudes.
Step-by-step explanation:
The question asks for the coordinates of the orthocenter of a triangle with given vertices on a coordinate plane. The orthocenter is the point where the altitudes of a triangle intersect. To find the orthocenter, one needs to find the equations of the altitudes and then find their intersection point.
Part A: For the triangle with vertices (0,0), (16,8), and (8,44), we will find the slopes of the sides opposite the vertices. The slopes of the altitudes will be the negative reciprocals of these slopes since the altitudes are perpendicular to the sides. Then, using point-slope form, we will write the equations of the altitudes by passing them through their respective vertices. The intersection point of these altitudes will be the orthocenter.
Part B: For the triangle with vertices (3,4), (8,9), and (2,15), a similar method will be employed. Calculate the slopes of the sides, find the negative reciprocals for the slopes of the altitudes, and use the point-slope form to get the equations of the altitudes. Their intersection will give the orthocenter's coordinates.