1. **Problem statement:** Given a right triangle ABC, the altitude relative to the hypotenuse divides it into two segments measuring 3.6 cm and 6.4 cm. Find the lengths of the legs (catetos) of the triangle. 2. **Relevant formulas and rules:** - In a right triangle, the altitude to the hypotenuse creates two smaller right triangles similar to the original. - The altitude $h$ satisfies $h^2 = p \times q$, where $p$ and $q$ are the segments into which the hypotenuse is divided. - Each leg $a$ and $b$ satisfies $a^2 = p \times c$ and $b^2 = q \times c$, where $c$ is the hypotenuse. 3. **Calculate the hypotenuse $c$:** $$c = p + q = 3.6 + 6.4 = 10$$ 4. **Calculate the altitude $h$:** $$h = \sqrt{p \times q} = \sqrt{3.6 \times 6.4} = \sqrt{23.04} = 4.8$$ 5. **Calculate the legs $a$ and $b$:** - For leg $a$ corresponding to segment $p$: $$a = \sqrt{p \times c} = \sqrt{3.6 \times 10} = \sqrt{36} = 6$$ - For leg $b$ corresponding to segment $q$: $$b = \sqrt{q \times c} = \sqrt{6.4 \times 10} = \sqrt{64} = 8$$ 6. **Answer:** The legs of the triangle measure $6$ cm and $8$ cm.