2. Let $A B C$ be a triangle with $\angle A=90^{\circ}, A B=1$, and $A C=2$. Let $\ell$ be a line through $A$ perpendicular to $B C$, and let the perpendicular bisectors of $A B$ and $A C$ meet $\ell$ at $E$ and $F$, respectively. Find the length of segment $E F$.
Answer: $\frac{3 \sqrt{5}}{4}$ Let $M, N$ be the midpoints of $A B$ and $A C$, respectively. Then we have $\angle E A B=$ $\angle A C B$ and $\angle E A C=\angle A B C$, so $A E M \sim C B A \Rightarrow A E=\frac{\sqrt{5}}{4}$ and $F A N \sim C B A \Rightarrow A F=\sqrt{5}$. Consequently, $E F=A F-A E=\frac{3 \sqrt{5}}{4}$.