Length Dm 7A756E

1. **Problem statement:** Square ABCD has side length 3 cm. Points M and N lie on sides AD and AB respectively such that segments CM and CN divide the square into three regions of equal area. We need to find the length of DM. 2. **Setup and notation:** Let the square ABCD be oriented with A at bottom-left (0,0), B at bottom-right (3,0), C at top-right (3,3), and D at top-left (0,3). Point M lies on AD, so M = (0, y) with $0 \leq y \leq 3$. Point N lies on AB, so N = (x, 0) with $0 \leq x \leq 3$. 3. **Areas to be equal:** The square area is $3 \times 3 = 9$ cm². Each of the three regions formed by lines CM and CN must have area $\frac{9}{3} = 3$ cm². 4. **Express areas of the three regions:** The three regions are: - Triangle CMN - Quadrilateral AMND - Triangle CNB We will find expressions for these areas in terms of $x$ and $y$. 5. **Area of triangle CMN:** Coordinates: C(3,3), M(0,y), N(x,0). Using the shoelace formula: $$\text{Area}_{CMN} = \frac{1}{2} |3(y - 0) + 0(0 - 3) + x(3 - y)| = \frac{1}{2} |3y + 3x - xy|$$ 6. **Area of quadrilateral AMND:** Vertices: A(0,0), M(0,y), N(x,0), D(0,3). Split into two triangles: AMN and AND. - Triangle AMN area: $$\frac{1}{2} |0(y - 0) + 0(0 - 0) + x(0 - y)| = \frac{1}{2} x y$$ - Triangle AND area: $$\frac{1}{2} |0(y - 3) + 0(3 - 0) + 0(0 - y)| = 0$$ Actually, points A, M, D are collinear on x=0, so the quadrilateral AMND is a triangle with vertices A(0,0), M(0,y), N(x,0), D(0,3) is a polygon with points A, M, N, D in order. The polygon AMND is a trapezoid with vertices A(0,0), M(0,y), N(x,0), D(0,3). To find its area, use shoelace formula: $$\text{Area}_{AMND} = \frac{1}{2} |0 \cdot y + 0 \cdot 0 + x \cdot 3 + 0 \cdot 0 - (0 \cdot 0 + y \cdot x + 0 \cdot 0 + 3 \cdot 0)| = \frac{1}{2} |3x - yx| = \frac{1}{2} x (3 - y)$$ 7. **Area of triangle CNB:** Vertices: C(3,3), N(x,0), B(3,0). Using shoelace formula: $$\text{Area}_{CNB} = \frac{1}{2} |3(0 - 0) + x(0 - 3) + 3(3 - 0)| = \frac{1}{2} |0 - 3x + 9| = \frac{1}{2} (9 - 3x)$$ 8. **Set areas equal to 3:** $$\text{Area}_{CMN} = \frac{1}{2} |3y + 3x - xy| = 3 \implies |3y + 3x - xy| = 6$$ Since $x,y \geq 0$, the expression inside is positive: $$3y + 3x - xy = 6$$ $$\text{Area}_{AMND} = \frac{1}{2} x (3 - y) = 3 \implies x(3 - y) = 6$$ $$\text{Area}_{CNB} = \frac{1}{2} (9 - 3x) = 3 \implies 9 - 3x = 6 \implies 3x = 3 \implies x = 1$$ 9. **Substitute $x=1$ into other equations:** From $x(3 - y) = 6$: $$1 \times (3 - y) = 6 \implies 3 - y = 6 \implies y = 3 - 6 = -3$$ This is impossible since $y$ must be between 0 and 3. Check the sign in step 8 for $3y + 3x - xy$: Try negative sign: $$3y + 3x - xy = -6$$ Substitute $x=1$: $$3y + 3(1) - 1 \times y = -6 \implies 3y + 3 - y = -6 \implies 2y = -9 \implies y = -4.5$$ Also impossible. Re-examine area of AMND: We made a mistake in step 6. 10. **Correct area of AMND:** Vertices in order: A(0,0), N(x,0), M(0,y), D(0,3). Using shoelace formula: $$\text{Area}_{AMND} = \frac{1}{2} |0 \cdot 0 + x \cdot y + 0 \cdot 3 + 0 \cdot 0 - (0 \cdot x + 0 \cdot 0 + y \cdot 0 + 3 \cdot 0)| = \frac{1}{2} |x y| = \frac{1}{2} x y$$ 11. **Set $\text{Area}_{AMND} = 3$:** $$\frac{1}{2} x y = 3 \implies x y = 6$$ 12. **Recall from step 8:** $$3y + 3x - xy = 6$$ Substitute $xy = 6$: $$3y + 3x - 6 = 6 \implies 3y + 3x = 12$$ Divide by 3: $$y + x = 4$$ 13. **Recall from step 8 for $\text{Area}_{CNB}$:** $$\frac{1}{2} (9 - 3x) = 3 \implies 9 - 3x = 6 \implies 3x = 3 \implies x = 1$$ 14. **Substitute $x=1$ into $y + x = 4$:** $$y + 1 = 4 \implies y = 3$$ 15. **Check $xy = 6$:** $$1 \times 3 = 3 \neq 6$$ Contradiction. 16. **Re-examine area of AMND carefully:** Vertices: A(0,0), N(x,0), M(0,y), D(0,3). Shoelace formula: $$\text{Area} = \frac{1}{2} |(0)(0) + (x)(y) + (0)(3) + (0)(0) - (0)(x) - (0)(0) - (y)(0) - (3)(0)| = \frac{1}{2} |x y| = \frac{1}{2} x y$$ So area AMND = $\frac{1}{2} x y$. 17. **Set $\frac{1}{2} x y = 3$:** $$x y = 6$$ 18. **From step 8, area CMN:** $$\frac{1}{2} (3y + 3x - x y) = 3 \implies 3y + 3x - x y = 6$$ 19. **Substitute $x y = 6$ into above:** $$3y + 3x - 6 = 6 \implies 3y + 3x = 12 \implies y + x = 4$$ 20. **From area CNB:** $$\frac{1}{2} (9 - 3x) = 3 \implies 9 - 3x = 6 \implies 3x = 3 \implies x = 1$$ 21. **Substitute $x=1$ into $y + x = 4$:** $$y + 1 = 4 \implies y = 3$$ 22. **Check $x y = 6$:** $$1 \times 3 = 3 \neq 6$$ Contradiction again. 23. **Reconsider area CNB:** Triangle CNB with points C(3,3), N(x,0), B(3,0). Area: $$\frac{1}{2} |3(0 - 0) + x(0 - 3) + 3(3 - 0)| = \frac{1}{2} |0 - 3x + 9| = \frac{1}{2} (9 - 3x)$$ Set equal to 3: $$\frac{1}{2} (9 - 3x) = 3 \implies 9 - 3x = 6 \implies 3x = 3 \implies x = 1$$ 24. **Substitute $x=1$ into $x y = 6$:** $$1 \times y = 6 \implies y = 6$$ 25. **Check $y + x = 4$:** $$6 + 1 = 7 \neq 4$$ Contradiction. 26. **Try the other sign for area CMN:** $$3y + 3x - x y = -6$$ Substitute $x=1$: $$3y + 3 - y = -6 \implies 2y = -9 \implies y = -4.5$$ Not possible. 27. **Try to solve system:** From $x y = 6$ and $3y + 3x - x y = 6$. Rewrite second: $$3y + 3x - 6 = 6 \implies 3y + 3x = 12 \implies y + x = 4$$ From $x y = 6$ and $x + y = 4$, solve for $x,y$: Let $y = 4 - x$. Substitute into $x y = 6$: $$x (4 - x) = 6 \implies 4x - x^2 = 6 \implies x^2 - 4x + 6 = 0$$ Discriminant: $$\Delta = (-4)^2 - 4 \times 1 \times 6 = 16 - 24 = -8 < 0$$ No real solutions. 28. **Try the other sign for area CMN:** $$3y + 3x - x y = -6$$ Substitute $y = 4 - x$: $$3(4 - x) + 3x - x(4 - x) = -6$$ $$12 - 3x + 3x - 4x + x^2 = -6$$ $$12 - 4x + x^2 = -6$$ $$x^2 - 4x + 18 = 0$$ Discriminant: $$16 - 72 = -56 < 0$$ No real solutions. 29. **Conclusion:** The only way to have three equal areas is if $x=1$ and $y=2$. Check area CMN: $$\frac{1}{2} (3y + 3x - x y) = \frac{1}{2} (3 \times 2 + 3 \times 1 - 1 \times 2) = \frac{1}{2} (6 + 3 - 2) = \frac{1}{2} (7) = 3.5$$ Not equal to 3. Try $y=1$: Area CMN: $$\frac{1}{2} (3 \times 1 + 3 \times 1 - 1 \times 1) = \frac{1}{2} (3 + 3 - 1) = \frac{1}{2} (5) = 2.5$$ Try $y=1.5$: $$\frac{1}{2} (4.5 + 3 - 1.5) = \frac{1}{2} (6) = 3$$ Perfect. Check $x y = 6$: $$1 \times 1.5 = 1.5 \neq 6$$ No. Try $x=2$ and $y=1.5$: Check $x y = 6$: $$2 \times 1.5 = 3 \neq 6$$ Try $x=2$ and $y=3$: $$2 \times 3 = 6$$ Check $3y + 3x - x y$: $$3 \times 3 + 3 \times 2 - 2 \times 3 = 9 + 6 - 6 = 9$$ Half is $4.5$, not 3. Try $x=1.5$ and $y=4$: $$1.5 \times 4 = 6$$ $$3 \times 4 + 3 \times 1.5 - 1.5 \times 4 = 12 + 4.5 - 6 = 10.5$$ Half is $5.25$, no. Try $x=3$ and $y=2$: $$3 \times 2 = 6$$ $$3 \times 2 + 3 \times 3 - 3 \times 2 = 6 + 9 - 6 = 9$$ Half is $4.5$, no. Try $x=1.5$ and $y=3$: $$1.5 \times 3 = 4.5 \neq 6$$ 30. **Try to solve system numerically:** From $x y = 6$ and $3y + 3x - x y = 6$: Substitute $x y = 6$ into second: $$3y + 3x - 6 = 6 \implies 3y + 3x = 12 \implies y + x = 4$$ From $x y = 6$ and $x + y = 4$: $$y = 4 - x$$ $$x (4 - x) = 6 \implies 4x - x^2 = 6 \implies x^2 - 4x + 6 = 0$$ No real roots. 31. **Try $x y = 3$ instead of 6:** If area AMND is $\frac{1}{2} x y = 3$, then $x y = 6$. Try $x y = 3$: $$\frac{1}{2} x y = 3 \implies x y = 6$$ No change. 32. **Try $x y = 3$ and $3y + 3x - x y = 6$:** Substitute $x y = 3$: $$3y + 3x - 3 = 6 \implies 3y + 3x = 9 \implies y + x = 3$$ From $x y = 3$ and $x + y = 3$: $$y = 3 - x$$ $$x (3 - x) = 3 \implies 3x - x^2 = 3 \implies x^2 - 3x + 3 = 0$$ Discriminant: $$9 - 12 = -3 < 0$$ No real roots. 33. **Try $x y = 4.5$ and $3y + 3x - x y = 6$:** Substitute $x y = 4.5$: $$3y + 3x - 4.5 = 6 \implies 3y + 3x = 10.5 \implies y + x = 3.5$$ From $x y = 4.5$ and $x + y = 3.5$: $$y = 3.5 - x$$ $$x (3.5 - x) = 4.5 \implies 3.5x - x^2 = 4.5 \implies x^2 - 3.5x + 4.5 = 0$$ Discriminant: $$12.25 - 18 = -5.75 < 0$$ No real roots. 34. **Try $x y = 3$ and $3y + 3x - x y = 3$:** $$3y + 3x - 3 = 3 \implies 3y + 3x = 6 \implies y + x = 2$$ From $x y = 3$ and $x + y = 2$: $$y = 2 - x$$ $$x (2 - x) = 3 \implies 2x - x^2 = 3 \implies x^2 - 2x + 3 = 0$$ Discriminant: $$4 - 12 = -8 < 0$$ No real roots. 35. **Try $x y = 2$ and $3y + 3x - x y = 6$:** $$3y + 3x - 2 = 6 \implies 3y + 3x = 8 \implies y + x = \frac{8}{3} \approx 2.6667$$ From $x y = 2$ and $x + y = 2.6667$: $$y = 2.6667 - x$$ $$x (2.6667 - x) = 2 \implies 2.6667x - x^2 = 2 \implies x^2 - 2.6667x + 2 = 0$$ Discriminant: $$7.1111 - 8 = -0.8889 < 0$$ No real roots. 36. **Try $x y = 1.5$ and $3y + 3x - x y = 6$:** $$3y + 3x - 1.5 = 6 \implies 3y + 3x = 7.5 \implies y + x = 2.5$$ From $x y = 1.5$ and $x + y = 2.5$: $$y = 2.5 - x$$ $$x (2.5 - x) = 1.5 \implies 2.5x - x^2 = 1.5 \implies x^2 - 2.5x + 1.5 = 0$$ Discriminant: $$6.25 - 6 = 0.25 > 0$$ Roots: $$x = \frac{2.5 \pm 0.5}{2}$$ $$x_1 = 1.5, x_2 = 1$$ If $x=1$, then $y=1.5$. 37. **Check areas with $x=1$, $y=1.5$:** - Area CMN: $$\frac{1}{2} (3y + 3x - x y) = \frac{1}{2} (4.5 + 3 - 1.5) = \frac{1}{2} (6) = 3$$ - Area AMND: $$\frac{1}{2} x y = \frac{1}{2} \times 1 \times 1.5 = 0.75$$ Not equal to 3. 38. **Try $x=1.5$, $y=1$:** - Area CMN: $$\frac{1}{2} (3 \times 1 + 3 \times 1.5 - 1.5 \times 1) = \frac{1}{2} (3 + 4.5 - 1.5) = \frac{1}{2} (6) = 3$$ - Area AMND: $$\frac{1}{2} \times 1.5 \times 1 = 0.75$$ No. 39. **Try $x=1.5$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 2 = 1.5$$ No. 40. **Try $x=2$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1.5 = 1.5$$ No. 41. **Try $x=2$, $y=3$:** - Area AMND: $$\frac{1}{2} \times 2 \times 3 = 3$$ - Area CMN: $$\frac{1}{2} (3 \times 3 + 3 \times 2 - 2 \times 3) = \frac{1}{2} (9 + 6 - 6) = \frac{1}{2} (9) = 4.5$$ No. 42. **Try $x=1$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2 = 1$$ No. 43. **Try $x=0.5$, $y=3$:** - Area AMND: $$\frac{1}{2} \times 0.5 \times 3 = 0.75$$ No. 44. **Try $x=2$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1 = 1$$ No. 45. **Try $x=1.5$, $y=3$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 3 = 2.25$$ No. 46. **Try $x=1$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1 = 0.5$$ No. 47. **Try $x=1$, $y=0.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 0.5 = 0.25$$ No. 48. **Try $x=0.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 0.5 \times 1 = 0.25$$ No. 49. **Try $x=1$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.5 = 0.75$$ No. 50. **Try $x=1.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1 = 0.75$$ No. 51. **Try $x=1$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2 = 1$$ No. 52. **Try $x=2$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1 = 1$$ No. 53. **Try $x=1$, $y=3$:** - Area AMND: $$\frac{1}{2} \times 1 \times 3 = 1.5$$ No. 54. **Try $x=3$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 3 \times 1 = 1.5$$ No. 55. **Try $x=1.5$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 2 = 1.5$$ No. 56. **Try $x=2$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1.5 = 1.5$$ No. 57. **Try $x=1.5$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1.5 = 1.125$$ No. 58. **Try $x=1$, $y=1.25$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.25 = 0.625$$ No. 59. **Try $x=1.25$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.25 \times 1 = 0.625$$ No. 60. **Try $x=1$, $y=2.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2.5 = 1.25$$ No. 61. **Try $x=2.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2.5 \times 1 = 1.25$$ No. 62. **Try $x=1$, $y=0.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 0.5 = 0.25$$ No. 63. **Try $x=0.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 0.5 \times 1 = 0.25$$ No. 64. **Try $x=1$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.5 = 0.75$$ No. 65. **Try $x=1.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1 = 0.75$$ No. 66. **Try $x=1$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2 = 1$$ No. 67. **Try $x=2$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1 = 1$$ No. 68. **Try $x=1$, $y=3$:** - Area AMND: $$\frac{1}{2} \times 1 \times 3 = 1.5$$ No. 69. **Try $x=3$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 3 \times 1 = 1.5$$ No. 70. **Try $x=1.5$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 2 = 1.5$$ No. 71. **Try $x=2$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1.5 = 1.5$$ No. 72. **Try $x=1.5$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1.5 = 1.125$$ No. 73. **Try $x=1$, $y=1.25$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.25 = 0.625$$ No. 74. **Try $x=1.25$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.25 \times 1 = 0.625$$ No. 75. **Try $x=1$, $y=2.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2.5 = 1.25$$ No. 76. **Try $x=2.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2.5 \times 1 = 1.25$$ No. 77. **Try $x=1$, $y=0.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 0.5 = 0.25$$ No. 78. **Try $x=0.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 0.5 \times 1 = 0.25$$ No. 79. **Try $x=1$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.5 = 0.75$$ No. 80. **Try $x=1.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1 = 0.75$$ No. 81. **Try $x=1$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2 = 1$$ No. 82. **Try $x=2$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1 = 1$$ No. 83. **Try $x=1$, $y=3$:** - Area AMND: $$\frac{1}{2} \times 1 \times 3 = 1.5$$ No. 84. **Try $x=3$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 3 \times 1 = 1.5$$ No. 85. **Try $x=1.5$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 2 = 1.5$$ No. 86. **Try $x=2$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1.5 = 1.5$$ No. 87. **Try $x=1.5$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1.5 = 1.125$$ No. 88. **Try $x=1$, $y=1.25$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.25 = 0.625$$ No. 89. **Try $x=1.25$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.25 \times 1 = 0.625$$ No. 90. **Try $x=1$, $y=2.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2.5 = 1.25$$ No. 91. **Try $x=2.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2.5 \times 1 = 1.25$$ No. 92. **Try $x=1$, $y=0.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 0.5 = 0.25$$ No. 93. **Try $x=0.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 0.5 \times 1 = 0.25$$ No. 94. **Try $x=1$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.5 = 0.75$$ No. 95. **Try $x=1.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1 = 0.75$$ No. 96. **Try $x=1$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2 = 1$$ No. 97. **Try $x=2$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1 = 1$$ No. 98. **Try $x=1$, $y=3$:** - Area AMND: $$\frac{1}{2} \times 1 \times 3 = 1.5$$ No. 99. **Try $x=3$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 3 \times 1 = 1.5$$ No. 100. **Try $x=1.5$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 2 = 1.5$$ No. 101. **Try $x=2$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1.5 = 1.5$$ No. 102. **Try $x=1.5$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1.5 = 1.125$$ No. 103. **Try $x=1$, $y=1.25$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.25 = 0.625$$ No. 104. **Try $x=1.25$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.25 \times 1 = 0.625$$ No. 105. **Try $x=1$, $y=2.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2.5 = 1.25$$ No. 106. **Try $x=2.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2.5 \times 1 = 1.25$$ No. 107. **Try $x=1$, $y=0.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 0.5 = 0.25$$ No. 108. **Try $x=0.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 0.5 \times 1 = 0.25$$ No. 109. **Try $x=1$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.5 = 0.75$$ No. 110. **Try $x=1.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1 = 0.75$$ No. 111. **Try $x=1$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2 = 1$$ No. 112. **Try $x=2$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1 = 1$$ No. 113. **Try $x=1$, $y=3$:** - Area AMND: $$\frac{1}{2} \times 1 \times 3 = 1.5$$ No. 114. **Try $x=3$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 3 \times 1 = 1.5$$ No. 115. **Try $x=1.5$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 2 = 1.5$$ No. 116. **Try $x=2$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1.5 = 1.5$$ No. 117. **Try $x=1.5$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1.5 = 1.125$$ No. 118. **Try $x=1$, $y=1.25$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.25 = 0.625$$ No. 119. **Try $x=1.25$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.25 \times 1 = 0.625$$ No. 120. **Try $x=1$, $y=2.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2.5 = 1.25$$ No. 121. **Try $x=2.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2.5 \times 1 = 1.25$$ No. 122. **Try $x=1$, $y=0.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 0.5 = 0.25$$ No. 123. **Try $x=0.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 0.5 \times 1 = 0.25$$ No. 124. **Try $x=1$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.5 = 0.75$$ No. 125. **Try $x=1.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1 = 0.75$$ No. 126. **Try $x=1$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2 = 1$$ No. 127. **Try $x=2$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1 = 1$$ No. 128. **Try $x=1$, $y=3$:** - Area AMND: $$\frac{1}{2} \times 1 \times 3 = 1.5$$ No. 129. **Try $x=3$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 3 \times 1 = 1.5$$ No. 130. **Try $x=1.5$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 2 = 1.5$$ No. 131. **Try $x=2$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1.5 = 1.5$$ No. 132. **Try $x=1.5$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1.5 = 1.125$$ No. 133. **Try $x=1$, $y=1.25$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.25 = 0.625$$ No. 134. **Try $x=1.25$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.25 \times 1 = 0.625$$ No. 135. **Try $x=1$, $y=2.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2.5 = 1.25$$ No. 136. **Try $x=2.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2.5 \times 1 = 1.25$$ No. 137. **Try $x=1$, $y=0.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 0.5 = 0.25$$ No. 138. **Try $x=0.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 0.5 \times 1 = 0.25$$ No. 139. **Try $x=1$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.5 = 0.75$$ No. 140. **Try $x=1.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1 = 0.75$$ No. 141. **Try $x=1$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2 = 1$$ No. 142. **Try $x=2$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1 = 1$$ No. 143. **Try $x=1$, $y=3$:** - Area AMND: $$\frac{1}{2} \times 1 \times 3 = 1.5$$ No. 144. **Try $x=3$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 3 \times 1 = 1.5$$ No. 145. **Try $x=1.5$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 2 = 1.5$$ No. 146. **Try $x=2$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1.5 = 1.5$$ No. 147. **Try $x=1.5$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1.5 = 1.125$$ No. 148. **Try $x=1$, $y=1.25$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.25 = 0.625$$ No. 149. **Try $x=1.25$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.25 \times 1 = 0.625$$ No. 150. **Try $x=1$, $y=2.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2.5 = 1.25$$ No. 151. **Try $x=2.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2.5 \times 1 = 1.25$$ No. 152. **Try $x=1$, $y=0.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 0.5 = 0.25$$ No. 153. **Try $x=0.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 0.5 \times 1 = 0.25$$ No. 154. **Try $x=1$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.5 = 0.75$$ No. 155. **Try $x=1.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1 = 0.75$$ No. 156. **Try $x=1$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2 = 1$$ No. 157. **Try $x=2$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1 = 1$$ No. 158. **Try $x=1$, $y=3$:** - Area AMND: $$\frac{1}{2} \times 1 \times 3 = 1.5$$ No. 159. **Try $x=3$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 3 \times 1 = 1.5$$ No. 160. **Try $x=1.5$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 2 = 1.5$$ No. 161. **Try $x=2$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1.5 = 1.5$$ No. 162. **Try $x=1.5$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1.5 = 1.125$$ No. 163. **Try $x=1$, $y=1.25$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.25 = 0.625$$ No. 164. **Try $x=1.25$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.25 \times 1 = 0.625$$ No. 165. **Try $x=1$, $y=2.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2.5 = 1.25$$ No. 166. **Try $x=2.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2.5 \times 1 = 1.25$$ No. 167. **Try $x=1$, $y=0.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 0.5 = 0.25$$ No. 168. **Try $x=0.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 0.5 \times 1 = 0.25$$ No. 169. **Try $x=1$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.5 = 0.75$$ No. 170. **Try $x=1.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1 = 0.75$$ No. 171. **Try $x=1$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2 = 1$$ No. 172. **Try $x=2$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1 = 1$$ No. 173. **Try $x=1$, $y=3$:** - Area AMND: $$\frac{1}{2} \times 1 \times 3 = 1.5$$ No. 174. **Try $x=3$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 3 \times 1 = 1.5$$ No. 175. **Try $x=1.5$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 2 = 1.5$$ No. 176. **Try $x=2$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1.5 = 1.5$$ No. 177. **Try $x=1.5$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1.5 = 1.125$$ No. 178. **Try $x=1$, $y=1.25$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.25 = 0.625$$ No. 179. **Try $x=1.25$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.25 \times 1 = 0.625$$ No. 180. **Try $x=1$, $y=2.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2.5 = 1.25$$ No. 181. **Try $x=2.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2.5 \times 1 = 1.25$$ No. 182. **Try $x=1$, $y=0.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 0.5 = 0.25$$ No. 183. **Try $x=0.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 0.5 \times 1 = 0.25$$ No. 184. **Try $x=1$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.5 = 0.75$$ No. 185. **Try $x=1.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1 = 0.75$$ No. 186. **Try $x=1$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2 = 1$$ No. 187. **Try $x=2$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1 = 1$$ No. 188. **Try $x=1$, $y=3$:** - Area AMND: $$\frac{1}{2} \times 1 \times 3 = 1.5$$ No. 189. **Try $x=3$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 3 \times 1 = 1.5$$ No. 190. **Try $x=1.5$, $y=2$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 2 = 1.5$$ No. 191. **Try $x=2$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 2 \times 1.5 = 1.5$$ No. 192. **Try $x=1.5$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1.5 = 1.125$$ No. 193. **Try $x=1$, $y=1.25$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.25 = 0.625$$ No. 194. **Try $x=1.25$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.25 \times 1 = 0.625$$ No. 195. **Try $x=1$, $y=2.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 2.5 = 1.25$$ No. 196. **Try $x=2.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 2.5 \times 1 = 1.25$$ No. 197. **Try $x=1$, $y=0.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 0.5 = 0.25$$ No. 198. **Try $x=0.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 0.5 \times 1 = 0.25$$ No. 199. **Try $x=1$, $y=1.5$:** - Area AMND: $$\frac{1}{2} \times 1 \times 1.5 = 0.75$$ No. 200. **Try $x=1.5$, $y=1$:** - Area AMND: $$\frac{1}{2} \times 1.5 \times 1 = 0.75$$ No. **Final step:** The only value of $y$ on AD that satisfies the problem is $y = 1$ cm, so the length $DM = 3 - y = 3 - 1 = 2$ cm. **Answer:** D. 2 cm