1. **State the problem:** We have two similar triangles sharing vertex S. We know side lengths 45, 15, and 16, and we need to find the missing side length $UV$ in the smaller triangle. 2. **Use the property of similar triangles:** Corresponding sides of similar triangles are proportional. This means: $$\frac{UV}{16} = \frac{15}{45}$$ 3. **Set up the proportion:** $$\frac{UV}{16} = \frac{15}{45}$$ 4. **Simplify the right side:** $$\frac{15}{45} = \frac{\cancel{15}}{\cancel{45}} = \frac{1}{3}$$ 5. **Rewrite the equation:** $$\frac{UV}{16} = \frac{1}{3}$$ 6. **Solve for $UV$ by cross-multiplying:** $$3 \times UV = 16 \times 1$$ 7. **Simplify:** $$3UV = 16$$ 8. **Divide both sides by 3:** $$\cancel{3}UV = \frac{16}{\cancel{3}}$$ 9. **Final answer:** $$UV = \frac{16}{3} \approx 5.33$$ So, the missing side length $UV$ is $\frac{16}{3}$ or approximately 5.33.