1. Problem 21 Part 1: Point C (-1, 2) divides segment AB in the ratio 3:4, with A(2, 5). Find coordinates of B. 2. Use the section formula: If point C divides AB in ratio m:n, then $$x_C = \frac{mx_B + nx_A}{m+n}, \quad y_C = \frac{my_B + ny_A}{m+n}$$ 3. Plug in values. Here, m=3, n=4, A=(2,5), C=(-1,2), and B=(x_B,y_B) unknown. $$-1 = \frac{3x_B + 4\times2}{7} \implies 3x_B +8 = -7 \implies 3x_B = -15 \implies x_B = -5$$ $$2 = \frac{3y_B + 4\times5}{7} \implies 3y_B +20 = 14 \implies 3y_B = -6 \implies y_B = -2$$ 4. So, coordinates of B are $(-5, -2)$. 5. Problem 21 Part 2: ABCD is a parallelogram with A(3,1), B(5,1), D(4,3), and C(a,b). Find a,b. 6. In a parallelogram, diagonals bisect each other. 7. Midpoint of AC = Midpoint of BD $$\text{Midpoint of AC} = \left(\frac{3+a}{2}, \frac{1+b}{2}\right)$$ $$\text{Midpoint of BD} = \left(\frac{5+4}{2}, \frac{1+3}{2}\right) = \left(\frac{9}{2}, 2\right)$$ 8. Equate midpoints: $$\frac{3+a}{2} = \frac{9}{2} \implies 3 + a = 9 \implies a = 6$$ $$\frac{1+b}{2} = 2 \implies 1 + b = 4 \implies b = 3$$ 9. Values are $a=6$, $b=3$. 10. Problem 22: In triangle ABC, D and E on AB and AC with DE || BC. Given $AD = x$, $DB = x - 2$, $AE = x + 2$, $EC = x - 1$, find $x$. 11. Because DE || BC, triangles ADE and ABC are similar. So ratios of corresponding sides are equal: $$\frac{AD}{AB} = \frac{AE}{AC}$$ 12. Calculate AB and AC: $$AB = AD + DB = x + (x - 2) = 2x - 2$$ $$AC = AE + EC = (x + 2) + (x - 1) = 2x + 1$$ 13. Substitute: $$\frac{x}{2x - 2} = \frac{x + 2}{2x + 1}$$ 14. Cross multiply: $$x(2x + 1) = (x + 2)(2x - 2)$$ $$2x^2 + x = 2x^2 - 2x + 4x -4$$ $$2x^2 + x = 2x^2 + 2x - 4$$ 15. Simplify by subtracting $2x^2$ from both sides: $$x = 2x - 4 \implies x - 2x = -4 \implies -x = -4 \implies x = 4$$ 16. Final answers: - Coordinates of B: $(-5, -2)$ - Coordinates of C: $(6, 3)$ - Value of $x$: 4