1. **State the problem:** We need to find the length $x$ in a right triangle where one angle is $30^\circ$ and one segment of the base is 9 cm. 2. **Identify the relevant formula:** In a right triangle, the side opposite the $30^\circ$ angle is half the hypotenuse. This is a property of a $30^\circ-60^\circ-90^\circ$ triangle. 3. **Analyze the triangle:** The side opposite the $30^\circ$ angle is 9 cm, so the hypotenuse is twice that length. 4. **Calculate the hypotenuse:** $$\text{Hypotenuse} = 2 \times 9 = 18 \text{ cm}$$ 5. **Find $x$:** Since the base is divided into two segments, one is 9 cm and the other is $x$, and the hypotenuse is 18 cm, the length $x$ is the remaining part of the base. 6. **Use the Pythagorean theorem or properties of the triangle:** In this case, the base is the side adjacent to the $30^\circ$ angle, which is $x + 9$ cm. The side adjacent to $30^\circ$ in a $30^\circ-60^\circ-90^\circ$ triangle is $\sqrt{3}$ times the shorter leg (opposite $30^\circ$). 7. **Calculate $x + 9$:** $$x + 9 = 9 \sqrt{3}$$ 8. **Solve for $x$:** $$x = 9 \sqrt{3} - 9 = 9(\sqrt{3} - 1)$$ **Final answer:** $$x = 9(\sqrt{3} - 1)$$