1. **State the problem:** We have two similar triangles ABC and DEF. We know the sides of triangle DEF: DE = 9 cm, EF = 6 cm, and the sides of triangle ABC: BC = 3 cm, AB = x cm. We need to find the value of $x$. 2. **Recall the property of similar triangles:** Corresponding sides of similar triangles are in proportion. This means: $$\frac{AB}{DE} = \frac{BC}{EF}$$ 3. **Substitute the known values:** $$\frac{x}{9} = \frac{3}{6}$$ 4. **Simplify the right side:** $$\frac{3}{6} = \frac{\cancel{3}}{\cancel{6}} = \frac{1}{2}$$ 5. **Set up the equation:** $$\frac{x}{9} = \frac{1}{2}$$ 6. **Solve for $x$ by cross-multiplying:** $$2 \times x = 9 \times 1$$ $$2x = 9$$ 7. **Divide both sides by 2:** $$x = \frac{9}{2}$$ 8. **Final answer:** $$x = 4.5$$ cm This means the length of side AB in triangle ABC is 4.5 cm.