In the diagram, what is the area of $\triangle ABC$? [asy]
size(5cm);
real xmax = 15; real ymax = 13; real xmin = -3; real ymin = -3;
pair a = (4, 9); pair b = (0, 0); pair c = (12, 0); draw(b--a--c);
draw((xmin, 0)--(xmax, 0), Arrow); label("$x$", (xmax, 0), E);
draw((0, ymin)--(0, ymax), Arrow); label("$y$", (0, ymax), N);
label("$B(0, 0)$", b, SW);
label("$C(12, 0)$", c, S);
label("$A(4, 9)$", a, N);
[/asy]
We think of $BC$ as the base of $\triangle ABC$. Its length is $12$.

Since the $y$-coordinate of $A$ is $9$, then the height of $\triangle ABC$ from base $BC$ is $9$.

Therefore, the area of $\triangle ABC$ is $\frac12 (12)(9) = \boxed{54}.$