To cross the river

Using one plank diagonally across the outside corner (forming an isosceles triangle), and the other perpendicular from the first plank to the inside corner, we can create a self-supporting bridge.

The distance we need to cover, from the outside corner to the inside corner, is $\sqrt{4m^2 + 4m^2} = \color{#D61F06}5.657m$

Measurements allow for a 0.05m overlap to be used to support the planks at each end.

Allowing $0.05m$ for support at each end of the planks, the first diagonal plank will effectively achieve a distance of $\dfrac{\color{#3D99F6}3.8m}{2} = \color{#20A900}1.9m$ from the outside corner (due to the symmetry of the isosceles triangle).

This leaves $\color{#D61F06}5.657m \color{#333333}- \color{#20A900}1.9m \color{#333333}= \color{#E81990}3.757m$ to cover with the remaining plank, which is certainly possible with a plank of $\color{#3D99F6}3.9m$ , even with an allowance at either end of the plank for support.

If the planks had negligible width, they would have to be greater than $\dfrac{2}{3}\sqrt{4m^2 + 4m^2} = 3.77123616632825m$ in length to be able to form a self-supporting bridge.