1. **State the problem:** We have two similar triangles, and we need to find the missing side length $x$ using proportions. 2. **Recall the rule for similar triangles:** Corresponding sides of similar triangles are proportional. If $\triangle ABC \sim \triangle DEF$, then $$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$$ 3. **Set up the proportion:** From the given triangles, the sides correspond as follows: - $AB = 15$ corresponds to $DE = 10$ - $BC = x$ corresponds to $EF = 6$ So the proportion is: $$\frac{15}{x} = \frac{10}{6}$$ 4. **Cross multiply:** $$15 \times 6 = 10 \times x$$ 5. **Write the equation:** $$90 = 10x$$ 6. **Solve for $x$ by dividing both sides by 10:** $$x = \frac{90}{10}$$ 7. **Simplify the fraction:** $$x = \cancel{\frac{90}{10}}{\frac{9 \times 10}{10}} = 9$$ **Final answer:** $$x = 9$$