Farmer Pythagoras has a field in the shape of a right triangle with legs $3$ and $4$. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$. The shortest distance from $S$ to the hypotenuse is $1$ unit. What fraction of the field is planted?

1. With legs $p,q$, hypotenuse $r=\sqrt{p^2+q^2}$, total area $T=\frac{pq}2$, and distance $d$ from square (side $x$) to the hypotenuse, one finds $x=\dfrac{2T-rd}{p+q}$.
2. Here $p=3,q=4,r=5,T=6,d=1$: $x=\dfrac{12-5}{7}=1$.
3. Planted fraction $=1-\dfrac{x^2}T=1-\dfrac16=\boxed{\dfrac56}$.