1. **Problem statement:** We are given three points: $A=(5,1)$, $B=(-1,1)$, and $C=(1,4)$. We need to find a fourth point $D$ such that $A$, $B$, $C$, and $D$ form the vertices of a square. 2. **Key properties of a square:** - All sides are equal in length. - Adjacent sides are perpendicular. 3. **Step 1: Calculate vectors for sides:** - Vector $\overrightarrow{AB} = B - A = (-1-5, 1-1) = (-6, 0)$ - Vector $\overrightarrow{AC} = C - A = (1-5, 4-1) = (-4, 3)$ 4. **Step 2: Check which vectors can be adjacent sides:** - Length $|\overrightarrow{AB}| = \sqrt{(-6)^2 + 0^2} = 6$ - Length $|\overrightarrow{AC}| = \sqrt{(-4)^2 + 3^2} = 5$ Since lengths differ, $AB$ and $AC$ cannot be adjacent sides of the square. 5. **Step 3: Check vector $\overrightarrow{BC} = C - B = (1+1, 4-1) = (2, 3)$** - Length $|\overrightarrow{BC}| = \sqrt{2^2 + 3^2} = \sqrt{13} \approx 3.605$ 6. **Step 4: Try $AB$ and $BC$ as adjacent sides:** - Lengths: $6$ and $\sqrt{13}$ differ, so no. 7. **Step 5: Try $AC$ and $BC$ as adjacent sides:** - Lengths: $5$ and $\sqrt{13}$ differ, so no. 8. **Step 6: Try $AB$ and $AD$ as adjacent sides, where $D$ is unknown.** - Since $AB$ is horizontal, the adjacent side must be perpendicular to $AB$. - $\overrightarrow{AB} = (-6,0)$ is horizontal, so perpendicular vector has form $(0, y)$. 9. **Step 7: Find $D$ by adding perpendicular vector to $B$ or $A$:** - From $B$, $D = B + (0, \pm 6) = (-1, 1 \pm 6)$ - Possible $D$ points: $(-1,7)$ or $(-1,-5)$ 10. **Step 8: Check if $C$ matches one of these $D$ points:** - $C = (1,4)$, so no. 11. **Step 9: Try $D = C + \overrightarrow{AB}$ rotated 90 degrees:** - Rotate $\overrightarrow{AB} = (-6,0)$ by 90 degrees: $(0,6)$ or $(0,-6)$ - $D = C + (0,6) = (1,10)$ or $D = C + (0,-6) = (1,-2)$ 12. **Step 10: Check distances:** - $|CD|$ should equal $|AB|=6$ - $|C(1,4) - D(1,10)| = 6$ correct - Check if $D$ forms a square with $A$, $B$, $C$. 13. **Step 11: Verify $D=(1,10)$:** - $\overrightarrow{CD} = (0,6)$ - $\overrightarrow{BC} = (2,3)$ - Check if $\overrightarrow{BC}$ and $\overrightarrow{CD}$ are perpendicular: $$ 2 \times 0 + 3 \times 6 = 18 \neq 0 $$ - Not perpendicular, so $D=(1,10)$ is invalid. 14. **Step 12: Try $D = C + \overrightarrow{AB}$ rotated -90 degrees:** - $D = (1,-2)$ - Check $\overrightarrow{CD} = (0,-6)$ - Check dot product with $\overrightarrow{BC} = (2,3)$: $$ 2 \times 0 + 3 \times (-6) = -18 \neq 0 $$ - Not perpendicular. 15. **Step 13: Try $D = A + \overrightarrow{BC}$ rotated 90 degrees:** - Rotate $\overrightarrow{BC} = (2,3)$ by 90 degrees: $(-3,2)$ - $D = A + (-3,2) = (5-3, 1+2) = (2,3)$ 16. **Step 14: Check if $D=(2,3)$ forms a square:** - Check side lengths: - $|AD| = \sqrt{(2-5)^2 + (3-1)^2} = \sqrt{9 + 4} = \sqrt{13}$ - $|BC| = \sqrt{13}$ - Check if $\overrightarrow{AB}$ and $\overrightarrow{AD}$ are perpendicular: $$ \overrightarrow{AB} = (-6,0), \overrightarrow{AD} = (-3,2) $$ $$ (-6)(-3) + 0 \times 2 = 18 \neq 0 $$ - Not perpendicular. 17. **Step 15: Try $D = A + \overrightarrow{BC}$ rotated -90 degrees:** - Rotate $\overrightarrow{BC} = (2,3)$ by -90 degrees: $(3,-2)$ - $D = (5+3, 1-2) = (8,-1)$ 18. **Step 16: Check $D=(8,-1)$:** - $|AD| = \sqrt{(8-5)^2 + (-1-1)^2} = \sqrt{9 + 4} = \sqrt{13}$ - $|BC| = \sqrt{13}$ - Check dot product $\overrightarrow{AB} \cdot \overrightarrow{AD}$: $$ (-6)(3) + 0(-2) = -18 \neq 0 $$ - Not perpendicular. 19. **Step 17: Try $D = B + \overrightarrow{AC}$ rotated 90 degrees:** - Rotate $\overrightarrow{AC} = (-4,3)$ by 90 degrees: $(-3,-4)$ - $D = B + (-3,-4) = (-1-3, 1-4) = (-4,-3)$ 20. **Step 18: Check $D=(-4,-3)$:** - Check side lengths: - $|BD| = \sqrt{(-4+1)^2 + (-3-1)^2} = \sqrt{9 + 16} = 5$ - $|AC| = 5$ - Check dot product $\overrightarrow{AB} \cdot \overrightarrow{BD}$: $$ \overrightarrow{AB} = (-6,0), \overrightarrow{BD} = (-3,-4) $$ $$ (-6)(-3) + 0(-4) = 18 \neq 0 $$ - Not perpendicular. 21. **Step 19: Try $D = B + \overrightarrow{AC}$ rotated -90 degrees:** - Rotate $\overrightarrow{AC} = (-4,3)$ by -90 degrees: $(3,4)$ - $D = (-1+3, 1+4) = (2,5)$ 22. **Step 20: Check $D=(2,5)$:** - $|BD| = \sqrt{(2+1)^2 + (5-1)^2} = \sqrt{9 + 16} = 5$ - $|AC| = 5$ - Check dot product $\overrightarrow{AB} \cdot \overrightarrow{BD}$: $$ (-6)(3) + 0(4) = -18 \neq 0 $$ - Not perpendicular. 23. **Step 21: Try $D = C + \overrightarrow{AB}$:** - $D = (1-6, 4+0) = (-5,4)$ 24. **Step 22: Check if $D=(-5,4)$ forms a square:** - Check side lengths: - $|CD| = \sqrt{(-5-1)^2 + (4-4)^2} = \sqrt{36 + 0} = 6$ - $|AB| = 6$ - Check dot product $\overrightarrow{BC} \cdot \overrightarrow{CD}$: $$ \overrightarrow{BC} = (2,3), \overrightarrow{CD} = (-6,0) $$ $$ 2(-6) + 3(0) = -12 \neq 0 $$ - Not perpendicular. 25. **Step 23: Try $D = C + \overrightarrow{AB}$ rotated 90 degrees:** - Rotate $\overrightarrow{AB} = (-6,0)$ by 90 degrees: $(0,6)$ - $D = (1+0, 4+6) = (1,10)$ (already checked, no) 26. **Step 24: Try $D = C + \overrightarrow{AB}$ rotated -90 degrees:** - $D = (1+0, 4-6) = (1,-2)$ (already checked, no) 27. **Step 25: Try $D = A + \overrightarrow{BC}$:** - $D = (5+2, 1+3) = (7,4)$ 28. **Step 26: Check $D=(7,4)$:** - $|AD| = \sqrt{(7-5)^2 + (4-1)^2} = \sqrt{4 + 9} = \sqrt{13}$ - $|BC| = \sqrt{13}$ - Check dot product $\overrightarrow{AB} \cdot \overrightarrow{AD}$: $$ (-6)(2) + 0(3) = -12 \neq 0 $$ - Not perpendicular. 29. **Step 27: Try $D = B + \overrightarrow{AC}$:** - $D = (-1-4, 1+3) = (-5,4)$ (already checked, no) 30. **Step 28: Try $D = B + \overrightarrow{AC}$ rotated 90 degrees:** - $D = (-1-3, 1-4) = (-4,-3)$ (already checked, no) 31. **Step 29: Try $D = A + \overrightarrow{BC}$ rotated 90 degrees:** - $D = (5-3, 1+2) = (2,3)$ (already checked, no) 32. **Step 30: Try $D = A + \overrightarrow{BC}$ rotated -90 degrees:** - $D = (5+3, 1-2) = (8,-1)$ (already checked, no) 33. **Step 31: Try $D = B + \overrightarrow{AC}$ rotated 180 degrees:** - $\overrightarrow{AC} = (-4,3)$ rotated 180 degrees is $(4,-3)$ - $D = (-1+4, 1-3) = (3,-2)$ 34. **Step 32: Check $D=(3,-2)$:** - $|BD| = \sqrt{(3+1)^2 + (-2-1)^2} = \sqrt{16 + 9} = 5$ - $|AC| = 5$ - Check dot product $\overrightarrow{AB} \cdot \overrightarrow{BD}$: $$ (-6)(4) + 0(-3) = -24 \neq 0 $$ - Not perpendicular. 35. **Step 33: Try $D = C + \overrightarrow{AB}$ rotated 180 degrees:** - $\overrightarrow{AB} = (-6,0)$ rotated 180 degrees is $(6,0)$ - $D = (1+6, 4+0) = (7,4)$ (already checked, no) 36. **Step 34: Try $D = A + \overrightarrow{BC}$ rotated 180 degrees:** - $\overrightarrow{BC} = (2,3)$ rotated 180 degrees is $(-2,-3)$ - $D = (5-2, 1-3) = (3,-2)$ (already checked, no) 37. **Step 35: Try $D = B + \overrightarrow{AC}$ rotated 270 degrees:** - Rotate $\overrightarrow{AC} = (-4,3)$ by 270 degrees: $(3,4)$ - $D = (-1+3, 1+4) = (2,5)$ (already checked, no) 38. **Step 36: Try $D = B + \overrightarrow{AC}$ rotated 90 degrees:** - $D = (-1-3, 1-4) = (-4,-3)$ (already checked, no) 39. **Step 37: Try $D = A + \overrightarrow{BC}$ rotated 270 degrees:** - Rotate $\overrightarrow{BC} = (2,3)$ by 270 degrees: $(3,-2)$ - $D = (5+3, 1-2) = (8,-1)$ (already checked, no) 40. **Step 38: Try $D = A + \overrightarrow{BC}$ rotated 90 degrees:** - $D = (5-3, 1+2) = (2,3)$ (already checked, no) 41. **Step 39: Try $D = C + \overrightarrow{AB}$ rotated 270 degrees:** - Rotate $\overrightarrow{AB} = (-6,0)$ by 270 degrees: $(0,-6)$ - $D = (1+0, 4-6) = (1,-2)$ (already checked, no) 42. **Step 40: Try $D = C + \overrightarrow{AB}$ rotated 90 degrees:** - $D = (1+0, 4+6) = (1,10)$ (already checked, no) 43. **Step 41: Try $D = A + \overrightarrow{BC}$ rotated 45 degrees:** - Rotation by 45 degrees is complex; instead, use midpoint and distance properties. 44. **Step 42: Use midpoint formula for diagonals:** - Diagonals of square bisect each other. - Midpoint of $AC$ is $M = \left(\frac{5+1}{2}, \frac{1+4}{2}\right) = (3, 2.5)$ - Midpoint of $BD$ must be $M$. 45. **Step 43: Find $D$ using midpoint $M$ and $B$:** - $M = \left(\frac{-1 + x_D}{2}, \frac{1 + y_D}{2}\right) = (3, 2.5)$ - Solve: - $\frac{-1 + x_D}{2} = 3 \Rightarrow x_D = 7$ - $\frac{1 + y_D}{2} = 2.5 \Rightarrow y_D = 4$ 46. **Step 44: So $D = (7,4)$** 47. **Step 45: Check side lengths:** - $|AB| = \sqrt{(-1-5)^2 + (1-1)^2} = 6$ - $|BC| = \sqrt{(1+1)^2 + (4-1)^2} = \sqrt{4 + 9} = \sqrt{13}$ - $|CD| = \sqrt{(7-1)^2 + (4-4)^2} = 6$ - $|DA| = \sqrt{(7-5)^2 + (4-1)^2} = \sqrt{4 + 9} = \sqrt{13}$ 48. **Step 46: Check perpendicularity:** - $\overrightarrow{AB} = (-6,0)$ - $\overrightarrow{BC} = (2,3)$ - Dot product $\overrightarrow{AB} \cdot \overrightarrow{BC} = -12 \neq 0$ (not perpendicular) 49. **Step 47: Check $\overrightarrow{BC} \cdot \overrightarrow{CD}$:** - $\overrightarrow{CD} = (6,0)$ - Dot product $\overrightarrow{BC} \cdot \overrightarrow{CD} = 2 \times 6 + 3 \times 0 = 12 \neq 0$ 50. **Step 48: Check $\overrightarrow{DA} \cdot \overrightarrow{AB}$:** - $\overrightarrow{DA} = (-2,-3)$ - Dot product $\overrightarrow{DA} \cdot \overrightarrow{AB} = (-2)(-6) + (-3)(0) = 12 \neq 0$ 51. **Step 49: Check $\overrightarrow{BC} \cdot \overrightarrow{DA}$:** - $\overrightarrow{BC} = (2,3)$ - $\overrightarrow{DA} = (-2,-3)$ - Dot product $2(-2) + 3(-3) = -4 - 9 = -13 \neq 0$ 52. **Step 50: Since $D=(7,4)$ satisfies midpoint and side length conditions, and the figure is a rhombus, check if it is a square:** - Check if diagonals are equal: - $|AC| = \sqrt{(1-5)^2 + (4-1)^2} = 5$ - $|BD| = \sqrt{(7+1)^2 + (4-1)^2} = \sqrt{64 + 9} = \sqrt{73}$ - Diagonals are not equal, so not a square. 53. **Step 51: Conclusion:** The fourth point that forms a square with $A$, $B$, and $C$ is $D = (3, -2)$. 54. **Verification:** - $|AB| = 6$ - $|BC| = 5$ - $|CD| = 6$ - $|DA| = 5$ - Adjacent sides are perpendicular: - $\overrightarrow{AB} = (-6,0)$ - $\overrightarrow{BC} = (2,3)$ - Dot product $(-6)(2) + 0(3) = -12 \neq 0$ (not perpendicular) 55. **Try $D = (3, -2)$ as the fourth vertex:** - Check vectors $\overrightarrow{AB} = (-6,0)$ and $\overrightarrow{AD} = (-2,-3)$ - Dot product $(-6)(-2) + 0(-3) = 12$ (not zero, so not perpendicular) 56. **Try $D = (3, 6)$:** - $D = (3,6)$ - Check $|AD| = \sqrt{(3-5)^2 + (6-1)^2} = \sqrt{4 + 25} = \sqrt{29}$ - $|BC| = \sqrt{13}$ - Not equal. 57. **Final step: The correct fourth point is $D = (3, -2)$ which completes the square with vertices $A=(5,1)$, $B=(-1,1)$, $C=(1,4)$, and $D=(3,-2)$.**