1. **State the problem:** We have two similar shapes. The left shape has sides 2 mm and $q$ mm, and the right shape has corresponding sides 1 mm and 4 mm. We need to find the missing length $q$. 2. **Formula for similar shapes:** Corresponding sides of similar shapes are proportional. This means: $$\frac{\text{side}_1}{\text{side}_2} = \frac{\text{corresponding side}_1}{\text{corresponding side}_2}$$ 3. **Set up the proportion:** Using the given sides, $$\frac{2}{q} = \frac{1}{4}$$ 4. **Solve for $q$:** Cross-multiply: $$2 \times 4 = 1 \times q$$ $$8 = q$$ 5. **Answer:** The missing length $q$ is 8 millimeters. This means the longer side in the left shape is 8 mm to keep the shapes similar.