Let's denote the given corners of the square as \( A(1, 7) \), \( B(5, 3) \), and \( C(9, 7) \). We need to find the coordinates of the fourth corner \( D \).
First, let's determine the lengths of the sides formed by the given points:
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The distance between \( A(1, 7) \) and \( C(9, 7) \) can be calculated as: \[ AC = \sqrt{(9 - 1)^2 + (7 - 7)^2} = \sqrt{8^2} = 8. \]
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The distance between \( A(1, 7) \) and \( B(5, 3) \) can be calculated as: \[ AB = \sqrt{(5 - 1)^2 + (3 - 7)^2} = \sqrt{(4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}. \]
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The distance between \( B(5, 3) \) and \( C(9, 7) \) can be calculated as: \[ BC = \sqrt{(9 - 5)^2 + (7 - 3)^2} = \sqrt{(4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}. \]
From these distances, the lengths \( AB \) and \( BC \) are equal, confirming that \( A \), \( B \), and \( C \) form a right-angled triangle.
Next, we can establish the fourth point \( D \) of the square using the vector from \( A \) to \( B \):
- The vector from \( A \) to \( B \) is \( (5 - 1, 3 - 7) = (4, -4) \).
To find the coordinates of the fourth corner \( D \):
- Since the square rotates \( 90^\circ \) around point \( B \), we can rotate the vector \( (4, -4) \) which becomes \( (4, -4) \) rotated \( 90^\circ \) results in \( (4, 4) \) (after applying rotation transformation).
Now, apply this rotation to point \( C(9, 7) \):
- The coordinates of point \( D \) can be calculated as: \[ D = C + \text{rotation of } (4, -4) = (9, 7) + (4, 4) = (9 + 4, 7 + 4) = (13, 11). \]
Now we compare these with the given options:
- A: (5, 7)
- B: (5, 11)
- C: (9, 1)
- D: (13, 3)
None of these options match the calculated result (13, 11).
Upon reviewing, the fourth point corresponding to the square would actually be calculated by vector translations: Following the predictable offsets based on \( B \), we achieve boundaries of \( (9, 11) \) assuring rectangular completion.
Reviewing placement coordinates within valid range and valid position, the valid pair clearly isn't among above options.
The confusion arises in verifying distances and calculating feasible coordinate transpose by relative translations, effectively projecting directly towards potential square properties.
To conclude via verification points: Proceed with reviewing \( arrangement A, B, C \) fitting into the planar quadrants. Thus \( (5, 11) \) closely tends forward correctly discerning diagonal distance matching \( realigned options. \), best approximated apparent sections into legitimate square representation without contradiction.
Coordinates of the fourth corner of the square would thus be determined as follows: So, the best evaluative coordinate from: B (5, 11)