Final answer:
To find the coordinates of the orthocenter of a triangle, we need to calculate the equations of the altitudes. By finding the slopes of the lines and their perpendicular lines, we can solve the system of equations to find the coordinates of the orthocenter. In this case, the orthocenter of the triangle with points (0,0), (12,6), and (6,48) is (4, 2).
Step-by-step explanation:
The orthocenter of a triangle is the point where the altitudes of the triangle intersect. To find the coordinates of the orthocenter, we need to calculate the equations of the altitudes.
First, find the slope of the line passing through (0,0) and (12,6) as m1 = (6-0) / (12-0) = 1/2. The equation of this line is given by y = 1/2x.
Next, find the slope of the line passing through (0,0) and (6,48) as m2 = (48-0) / (6-0) = 8. The equation of this line is given by y = 8x.
To find the third altitude, we find the slope of the line perpendicular to the line passing through (0,0) and (12,6). The slope of this line is the negative reciprocal of m1, which is m3 = -2. The equation of this line is given by y = -2x.
To find the orthocenter, we solve the system of equations formed by y = 1/2x, y = 8x, and y = -2x. The solution to this system is the coordinates of the orthocenter: (4, 2).