1. **State the problem:** We have two pairs of similar polygons. For each pair, we know the scale factor between the smaller and larger polygon and need to find the missing side length $x$ in the larger polygon. 2. **Formula and rules:** For similar polygons, corresponding sides are proportional. The scale factor $k$ is the ratio of a side length in the larger polygon to the corresponding side length in the smaller polygon: $$k = \frac{\text{side length in larger polygon}}{\text{side length in smaller polygon}}$$ To find the missing side length $x$ in the larger polygon: $$x = k \times \text{corresponding side length in smaller polygon}$$ 3. **First pair (quadrilaterals):** - Smaller polygon sides: 10, 14, 14, 3 - Larger polygon sides: 40, 56, 56, $x$ Calculate scale factor $k$ using known sides: $$k = \frac{40}{10} = 4$$ Check with another pair: $$k = \frac{56}{14} = 4$$ Scale factor is consistent: $k=4$ Find missing side $x$: $$x = 4 \times 3 = 12$$ 4. **Second pair (triangles):** - Smaller triangle sides: 9, 10, $x$ - Larger triangle sides: 54, 60, 30 Calculate scale factor $k$ using known sides: $$k = \frac{54}{9} = 6$$ Check with another pair: $$k = \frac{60}{10} = 6$$ Scale factor is consistent: $k=6$ Find missing side $x$ in smaller triangle using the corresponding side 30 in the larger triangle: $$x = \frac{30}{6} = 5$$ **Final answers:** - Missing side length in the larger quadrilateral: $12$ - Missing side length in the smaller triangle: $5$