1. **State the problem:** Solve the equation $$\frac{x}{9} = \frac{5}{x}$$ for $x$. 2. **Use the cross-multiplication rule:** When two fractions are equal, their cross products are equal. So, $$x \cdot x = 9 \cdot 5$$ 3. **Write the equation:** $$x^2 = 45$$ 4. **Solve for $x$ by taking the square root of both sides:** $$x = \pm \sqrt{45}$$ 5. **Simplify the square root:** $$\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}$$ 6. **Final answer:** $$x = \pm 3\sqrt{5}$$ This means $x$ can be either $3\sqrt{5}$ or $-3\sqrt{5}$. Note: Since the problem context involves a triangle side length $x$, which must be positive, we take the positive root: $$x = 3\sqrt{5}$$