1. **State the problem:** We have two similar trapezoidal shapes A and B. Shape A has sides: top = 4 cm, height = 12 cm, bottom = 7 cm. Shape B has sides: top = 3 cm, height = 9 cm, bottom = w cm (unknown). We need to find: a) The scale factor from shape A to shape B. b) The value of w. 2. **Formula and rules:** For similar shapes, corresponding sides are proportional. The scale factor $k$ from shape A to shape B is: $$k = \frac{\text{side in B}}{\text{corresponding side in A}}$$ All corresponding sides satisfy: $$\frac{\text{side in B}}{\text{side in A}} = k$$ 3. **Calculate the scale factor using the top sides:** $$k = \frac{3}{4}$$ 4. **Check scale factor with heights:** $$\frac{9}{12} = \frac{3}{4}$$ This confirms the scale factor is $\frac{3}{4}$. 5. **Find $w$ using the bottom sides:** Since the shapes are similar, the bottom sides satisfy: $$\frac{w}{7} = \frac{3}{4}$$ 6. **Solve for $w$:** $$w = 7 \times \frac{3}{4}$$ $$w = \frac{7 \times 3}{4} = \frac{21}{4}$$ 7. **Simplify the fraction:** $\frac{21}{4}$ is already in simplest form. **Final answers:** a) Scale factor from shape A to B is $\frac{3}{4}$. b) The value of $w$ is $\frac{21}{4}$.