2. Suppose $A B C$ is a scalene right triangle, and $P$ is the point on hypotenuse $\overline{A C}$ such that $\angle A B P=45^{\circ}$. Given that $A P=1$ and $C P=2$, compute the area of $A B C$.
Answer: $\frac{9}{5}$
Solution: Notice that $\overline{B P}$ bisects the right angle at $B$. Thus, we write $A B=2 x$, $B C=x$. By the Pythagorean theorem, $5 x^{2}=9$, from which the area $\frac{1}{2}(x)(2 x)=x^{2}=$ $\frac{9}{5}$.