.
ABC has vertices A(0, 6), B(4, 6), and C(1, 3). Sketch a graph of ABC and use it to find the orthocenter of ABC.
Then list the steps you took to find the orthocenter, including any necessary points or slopes you had to derive.
~~~~~~~~~~~~~~~~~~~~~~~~~~
The orthocenter is the point where the three altitudes of a triangle intersect.
Notice that the side AB is horizontal: it lies on the straight line y = 6.
Therefore, the altitude drawn to AB is VERTICAL line, and since it is going through the point C(1,3), its equation is
x = 1. (1)
Next, the side AC has the slope m = = = = -3.
Hence, the perpendicular to AC has the slope .
Thus, the altitude to the side AC, passing through the vertex B = (4,6), has the equation
y-6 = (2)
Equations (1) and (2) have a solution (x,y) = (1,5).
It is the orthocenter.
Answer. The orthocenter is the point (x,y) = (1,5).
--------------
P.S. You do not need the equation of the third altitude.
It is quite enough to have the equations of two altitudes and to find their common solution, which represents the common intersection point.
The third altitude will pass through this point, since the altitudes are concurrent in any triangle.