1. **Problem 5(a):** Find the ratio of PN to MN in simplest form given triangles ABC and MNP are similar. 2. Since triangles ABC and MNP are similar, corresponding sides are proportional. 3. Given AB = 12, BC = 18, MN = 8, and NP = x. 4. The ratio of corresponding sides AB to MN is $ \frac{12}{8} = \frac{3}{2} $. 5. The ratio of PN to MN corresponds to BC to AB, so $ \frac{PN}{MN} = \frac{BC}{AB} = \frac{18}{12} = \frac{3}{2} $. 6. Therefore, the ratio of PN to MN in simplest form is $ \boxed{\frac{3}{2}} $. 7. **Problem 5(b):** Find the value of $ x $. 8. Using similarity, the ratios of corresponding sides are equal: $$\frac{BC}{NP} = \frac{AB}{MN}$$ 9. Substitute known values: $$\frac{18}{x} = \frac{12}{8}$$ 10. Simplify the right side: $$\frac{18}{x} = \frac{3}{2}$$ 11. Cross-multiply: $$3x = 2 \times 18$$ 12. Calculate right side: $$3x = 36$$ 13. Divide both sides by 3: $$x = \frac{36}{3}$$ 14. Simplify: $$x = 12$$ 15. **Problem 6(a):** Dilate segment AB by scale factor 3 about the origin. 16. Original points: $A(-2, -1), B(3, -2)$. 17. Dilation formula about origin: $ (x, y) \to (3x, 3y) $. 18. Calculate A': $$A' = (3 \times -2, 3 \times -1) = (-6, -3)$$ 19. Calculate B': $$B' = (3 \times 3, 3 \times -2) = (9, -6)$$ 20. **Problem 6(b):** Translate A'B' two units left and seven units up. 21. Translation formula: $ (x, y) \to (x - 2, y + 7) $. 22. Calculate A'': $$A'' = (-6 - 2, -3 + 7) = (-8, 4)$$ 23. Calculate B'': $$B'' = (9 - 2, -6 + 7) = (7, 1)$$ **Final answers:** - Ratio $ \frac{PN}{MN} = \frac{3}{2} $ - $ x = 12 $ - Coordinates after dilation: $ A'(-6, -3), B'(9, -6) $ - Coordinates after translation: $ A''(-8, 4), B''(7, 1) $