smc

Let $D$ be the foot of the altitude from $A$ to $\overline{BC}$ extended past $B$. Let $AD=x$ and $BD=y$. Using the Pythagorean Theorem, we obtain the equations $$\begin{array}{r}{x}^2+y^2=1(1)\\ x^2+y^2+2ya+a^2=4a^2(2)\\ x^2+y^2+4ya+4a^2=4(3)\end{array}$$ Subtracting $(1)$ equation from $(2)$ and $(3)$, we get $$\begin{array}{r}2ya+a^2=4a^2-1(4)\\ 4ya+4a^2=3(5)\end{array}$$ Then, subtracting $2\times (4)$ from $(5)$ and rearranging, we get $10a^2=5$, so $BC=2a=\sqrt{2}\Rightarrow \boxed{\sqrt{2}}$