Confused about the “height” to a slanted base (area of a triangle)

I’m stuck on finding the area of a triangle when the base isn’t horizontal or vertical. I know the area formula is 1/2 × base × height, but I keep tripping over what the “height” actually is when the base is slanted.

Concrete example: A(1,1), B(7,4), C(4,8). If I pick AB as the base, I can get its length (sqrt(45)). The slope of AB is 1/2, so I wrote the line as y = 1 + 0.5(x − 1). Then at x = 4, the line’s y-value is 2.5, so I tried using 8 − 2.5 = 5.5 as the height (vertical difference). I also tried a simpler shortcut using 8 − 4 = 4 as the height (difference in y between C and B), which I’m pretty sure is wrong, and I get different areas depending on which one I pick.

I know the height is supposed to be the perpendicular distance from C to line AB, not just a vertical or horizontal gap. I’m just not sure how to compute that perpendicular distance cleanly from two points on the base. Could someone show me the correct way to get the height from C to AB in this example, and point out where my reasoning goes off? I want to understand how to do this properly when the base is slanted.