1. **State the problem:** We have two similar quadrilaterals. The side lengths of the larger one are 40 mm and 60 mm, and the smaller one has a side length of 42 mm and a missing side length $w$. We need to find $w$. 2. **Recall the property of similar figures:** Corresponding sides of similar figures are proportional. This means the ratio of one pair of corresponding sides equals the ratio of another pair. 3. **Set up the proportion:** The side of length 60 mm corresponds to 42 mm, and the side of length 40 mm corresponds to $w$. So, $$\frac{w}{40} = \frac{42}{60}$$ 4. **Solve for $w$:** Multiply both sides by 40: $$w = 40 \times \frac{42}{60}$$ 5. **Simplify the fraction:** $$\frac{42}{60} = \frac{7}{10}$$ So, $$w = 40 \times \frac{7}{10}$$ 6. **Calculate $w$:** $$w = 40 \times 0.7 = 28$$ 7. **Final answer:** The missing length $w$ is **28 millimeters**.