AMC12 2009 12B, Problem 9

Triangle $ABC$ has vertices $A = (3,0)$, $B = (0,3)$, and $C$, where $C$ is on the line $x + y = 7$.  What is the area of $\triangle ABC$?

Solution: Because the line $x + y = 7$ is parallel to $\overline {AB}$, the area of $\triangle ABC$ is independent of the location of $C$ on that line.  Therefore it may be assumed that $C$ is $(7,0)$.  In that case the triangle has base $AC = 4$ and altitude $3$, so its area is $\frac 12 \cdot 4 \cdot 3 = \fbox{6}$.

Answer: 6