1. **Problem statement:** We have a right triangle ABC with right angle at A, where AB = 6 cm and AC = 8 cm. AM is the altitude from A to hypotenuse BC, and BM = MC, so M is the midpoint of BC. We need to find the length of AM, denoted as $x$. 2. **Formula and rules:** In a right triangle, the altitude to the hypotenuse satisfies the relation $$AM = \frac{AB \times AC}{BC}$$ 3. **Calculate BC:** Using the Pythagorean theorem, $$BC = \sqrt{AB^2 + AC^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$$ 4. **Calculate AM:** Substitute values into the altitude formula, $$AM = \frac{6 \times 8}{10} = \frac{48}{10}$$ 5. **Simplify fraction:** $$AM = \frac{\cancel{48}}{\cancel{10}} = 4.8$$ 6. **Interpretation:** The length of the altitude $x = AM$ is 4.8 cm. **Final answer:** $$x = 4.8$$