Answer: B

Small Hint:

Put $A$ at the origin and represent $B$ and $C$ by vectors

Big Hint:

Write direction vectors for the two medians and set their dot product to zero

Solution:

Let $A=\mathbf0,$ $B=\mathbf b,$ and $C=\mathbf c,$ with $|\mathbf b|=\ell,$ $|\mathbf c|=6,$ and $|\mathbf c-\mathbf b|=7.$ The median directions are $\frac{\mathbf b+\mathbf c}{2}$ from $A$ and $\frac{\mathbf c}{2}-\mathbf b$ from $B.$ Their dot product is zero, so $$(\mathbf b+\mathbf c)\mathbin{\cdot}(\mathbf c-2\mathbf b)=0.$$ Using $\mathbf b\mathbin{\cdot}\mathbf c=\frac{\ell^2+36-49}{2},$ this simplifies to $\ell^2=17.$ Hence $AB=\sqrt{17}.$

Therefore, the correct answer is B.