1.1. The lengths of the sides of a right-angled triangle form an arithmetic progression with a common difference of 1 cm. Find the length of the hypotenuse.
1.1. Let $c$ be the length of the hypotenuse. Then the lengths of the legs are $c-1$ and $c-2$. We have

$$
(c-1)^{2}+(c-2)^{2}=c^{2}, \text { or } c^{2}-6 c+5=0
$$

from which $c=5$ (cm) (the second root of the equation does not satisfy the condition).

Answer: 5 cm.