1. **Problem statement:** Two similar hexagons have corresponding sides of 2 cm and 5 cm. (a) Find the ratio of their areas. (b) The area of the larger hexagon is 150 cm^2. Find the area of the smaller hexagon. 2. **Formula and rules:** - For similar polygons, the ratio of their areas is the square of the ratio of their corresponding sides. - If the ratio of sides is $\frac{a}{b}$, then the ratio of areas is $\left(\frac{a}{b}\right)^2$. 3. **Step (a) Find the ratio of their areas:** - Ratio of sides = $\frac{2}{5}$ - Ratio of areas = $\left(\frac{2}{5}\right)^2 = \frac{4}{25}$ 4. **Step (b) Find the area of the smaller hexagon:** - Let area of smaller hexagon = $A_s$ - Given area of larger hexagon = $A_l = 150$ cm$^2$ - Using ratio of areas: $\frac{A_s}{A_l} = \frac{4}{25}$ - Solve for $A_s$: $$A_s = A_l \times \frac{4}{25} = 150 \times \frac{4}{25} = 150 \times 0.16 = 24$$ **Final answers:** - (a) Ratio of areas = $\frac{4}{25}$ - (b) Area of smaller hexagon = 24 cm$^2$