1. **Stating the problem:** We have a right triangle $\triangle ABC$ with right angle at $A$. The base $CA = 5.8$ cm and the height $AB = 4.5$ cm. Point $H$ lies on the hypotenuse $CB$ such that $\angle H = 90^\circ$ and $\angle A = 90^\circ$. We need to find the length $x = HA$. 2. **Understanding the problem:** Since $\angle A$ is right, $AB$ and $AC$ are perpendicular. Point $H$ is the foot of the perpendicular from $A$ to hypotenuse $CB$. So $HA$ is the altitude from the right angle vertex to the hypotenuse. 3. **Formula used:** In a right triangle, the altitude $h$ from the right angle vertex to the hypotenuse satisfies: $$h = \frac{AB \times AC}{CB}$$ where $CB$ is the hypotenuse. 4. **Calculate the hypotenuse $CB$ using Pythagoras theorem:** $$CB = \sqrt{AB^2 + AC^2} = \sqrt{4.5^2 + 5.8^2} = \sqrt{20.25 + 33.64} = \sqrt{53.89}$$ 5. **Simplify $CB$:** $$CB = 7.34 \text{ cm (approx)}$$ 6. **Calculate the altitude $HA = x$:** $$x = \frac{AB \times AC}{CB} = \frac{4.5 \times 5.8}{7.34}$$ 7. **Intermediate step showing cancellation:** $$x = \frac{\cancel{4.5} \times 5.8}{\cancel{7.34}} \approx \frac{26.1}{7.34}$$ 8. **Final calculation:** $$x \approx 3.56 \text{ cm}$$ **Answer:** The length $x = HA$ is approximately 3.56 cm.