What is the rectangle side length?

Label some more points as follows:

By the Pythagorean Theorem on $\triangle GFI$ , $GI = \sqrt{3^2 + 4^2} = 5$ .

$\triangle GFI \sim \triangle GHI$ by AA similarity, so $\cfrac{EH}{EG} = \cfrac{FI}{GI}$ , or $\cfrac{AB}{4} = \cfrac{4}{5}$ , so $AB = \boxed{3.2}$ .

Let $EF$ be perpendicular to $BC$ and label the $3$ - $4$ - $5$ right triangle as $GHI$ as shown. We can see that $\triangle EFI$ and $\triangle GHI$ are similar. Then we have $\dfrac {EF}{EI} = \dfrac {GH}{GI}$ . $\implies AB = EF = \dfrac {GH}{GI} \times EI = \dfrac 45 \times 4 = \boxed{3.2}$ .

Answer = (4/5) x (3 + 1) = 3.2