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The points A (1, 5), B (5, 5), and C (5, 1) are three corners of a square ABCD.

What are the coordinates of D, the fourth corner?

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To find the coordinates of the fourth corner [tex]\( D \)[/tex] of the square [tex]\( ABCD \)[/tex], follow these steps precisely:

1. Identify the Given Points and Their Coordinates:
- Point [tex]\( A \)[/tex] has coordinates [tex]\( (1, 5) \)[/tex].
- Point [tex]\( B \)[/tex] has coordinates [tex]\( (5, 5) \)[/tex].
- Point [tex]\( C \)[/tex] has coordinates [tex]\( (5, 1) \)[/tex].

2. Understand the Arrangement of a Square:
In a square, all sides are equal, and the vertices are connected such that the angles between any two sides are right angles (90 degrees).

3. Recognize the Relationship Between Points:
- Since [tex]\( AB \)[/tex] and [tex]\( CD \)[/tex] are opposite sides of the square: [tex]\( AB \parallel CD \)[/tex].
- Since [tex]\( BC \)[/tex] and [tex]\( AD \)[/tex] are opposite sides of the square: [tex]\( BC \parallel AD \)[/tex].
- Each side of the square shares the same length.

4. Determine the Missing Point [tex]\( D \)[/tex]:
- For [tex]\( A \)[/tex] to [tex]\( B \)[/tex], the x-coordinate changes from 1 to 5, while the y-coordinate remains the same (5).
- For [tex]\( B \)[/tex] to [tex]\( C \)[/tex], the x-coordinate remains the same (5) while the y-coordinate changes from 5 to 1.

Utilize these changes to ascertain the coordinates of [tex]\( D \)[/tex].

5. Calculate Coordinates of [tex]\( D \)[/tex]:
- Moving from [tex]\( C \)[/tex] to [tex]\( D \)[/tex]: Since [tex]\( C = (5, 1) \)[/tex], we take the x-coordinate from [tex]\( A \)[/tex] and the y-coordinate from [tex]\( A \)[/tex].
- [tex]\( D \)[/tex] will involve a horizontal and vertical change opposite to the change that occurred from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] and [tex]\( B \)[/tex] to [tex]\( C \)[/tex], respectively.

Thus,
[tex]\[ D_x = A[0] + (C[0] - B[0]) = 1 + (5 - 5) = 1, \][/tex]
[tex]\[ D_y = A[1] + (C[1] - B[1]) = 5 + (1 - 5) = 1. \][/tex]

6. Conclusion:
The coordinates of the fourth corner [tex]\( D \)[/tex] are [tex]\( (1, 1) \)[/tex].

So, the detailed steps reveal the coordinates of point [tex]\( D \)[/tex], completing the square [tex]\( ABCD \)[/tex] are [tex]\( \boxed{(1, 1)} \)[/tex].
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