1. **Problem 1: Areas of Similar Triangles** We have two similar triangles A and B. - Base of A = 7 cm - Base of B = 14 cm - Area of A = 20 cm² We need to find the area of triangle B. 2. **Formula and Rules:** - For similar triangles, the ratio of their areas is the square of the ratio of their corresponding sides. - If $\frac{b_B}{b_A}$ is the ratio of bases, then $$\frac{\text{Area}_B}{\text{Area}_A} = \left(\frac{b_B}{b_A}\right)^2$$ 3. **Calculate the ratio of bases:** $$\frac{b_B}{b_A} = \frac{14}{7} = 2$$ 4. **Calculate the area of B:** $$\frac{\text{Area}_B}{20} = 2^2 = 4$$ $$\text{Area}_B = 20 \times 4 = 80 \text{ cm}^2$$ --- 5. **Problem 2: Volumes of Similar Pentagonal Prisms** We have two similar pentagonal prisms A and B. - Base width of A = 3 cm - Base width of B = 9 cm - Volume of A = 15 cm³ We need to find the volume of prism B. 6. **Formula and Rules:** - For similar 3D shapes, the ratio of their volumes is the cube of the ratio of their corresponding linear dimensions. - If $\frac{w_B}{w_A}$ is the ratio of base widths, then $$\frac{\text{Volume}_B}{\text{Volume}_A} = \left(\frac{w_B}{w_A}\right)^3$$ 7. **Calculate the ratio of base widths:** $$\frac{w_B}{w_A} = \frac{9}{3} = 3$$ 8. **Calculate the volume of B:** $$\frac{\text{Volume}_B}{15} = 3^3 = 27$$ $$\text{Volume}_B = 15 \times 27 = 405 \text{ cm}^3$$ **Final answers:** - Area of triangle B = 80 cm² - Volume of prism B = 405 cm³