1. **State the problem:** We have two similar triangles. The sides of the smaller triangle are 4 in and 13 in, and the corresponding sides of the larger triangle are $x$ and 22.75 in. We need to find the missing side length $x$ of the larger triangle. 2. **Formula and rule:** For similar triangles, corresponding sides are proportional. This means: $$\frac{x}{4} = \frac{22.75}{13}$$ 3. **Solve for $x$:** Multiply both sides by 4 to isolate $x$: $$x = 4 \times \frac{22.75}{13}$$ 4. **Calculate the value:** $$x = \frac{4 \times 22.75}{13} = \frac{91}{13}$$ 5. **Simplify the fraction:** $$x = 7 \cancel{\times 13} \div \cancel{13} = 7$$ 6. **Final answer:** The missing side length $x$ is **7 inches**.