Write an equation of the altitude from C to line 
segment AB. The coordinates of C are (4,12), 
A (2,1), and B (16,3). 

Here is the triangle.  
A is the lower left vertex where the red line
segment AC meets the blue line segment AB.

B is the lower right vertex where the dark blue line
segment AB meets the green line sengment BC

C is the upper vertex where the red line
segment AC meets the green line segment BC




Now let's draw in the altitude from vertex C 
(the upper vertex) to the dark blue line segment AB.
I'll draw it in light blue.



Let's find the slope of the dark blue line 
segment AB by using the slope formula:

     y2 - y1
m = —————————
     x2 - x1

where (x1, y1) = A(2, 1) and (x2, y2) = B(16, 3)

     (3) - (1)      2     1
m = ——————————— = ———— = ——— 
    (16) - (2)     14     7 

An altitude is perpendicular to the side it's drawn to.
So the slope of the altitude (light blue line segment) is
the reciprocal of 1/7 with the sign changed.  That would
be -7/1 or -7.

We know that a point on the altitude (light blue line) is
the vertex C(4,12), so we use (x1, y1) = (4, 12)

Now substitute in the point slope formula:

   y - y1 = m(x - x1)

   y - 12 = 7(x - 4)

   y - 12 = 7x - 28

        y = 7x - 16

That's the equation of the blue line segment which
is the light blue line.  However, this is the 
equation of the ENTIRE LINE as we see here, i.e., 
the EXTENDED altitude, not just the altitude,
which is only a line segment, as you see below.
I suppose that's what you wanted.



Edwin