1. The problem describes a right triangle with legs along the axes: one leg on the x-axis of length $3$, and the other leg on the y-axis of length $4$. 2. We are asked to find the length of the hypotenuse $l$, which connects points $(0,4)$ and $(3,0)$. 3. Using the Pythagorean theorem: $$x^2 + y^2 = l^2$$ 4. Substitute the known leg lengths: $$3^2 + 4^2 = l^2$$ 5. Compute the squares: $$9 + 16 = l^2$$ 6. Add the values: $$25 = l^2$$ 7. Take the positive square root to find $l$: $$l = \sqrt{25} = 5$$ 8. Therefore, the length of the hypotenuse is $5$ units. This shows how the Pythagorean theorem applies to find the length of the hypotenuse in a right triangle given the legs.