To find the coordinates of the fourth corner of the square given the three corners, we can use the properties of a square. The corners you provided are:
- \( A(1, 7) \)
- \( B(5, 3) \)
- \( C(9, 7) \)
Let's denote the fourth corner as \( D(x, y) \).
Steps to find D:
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Calculate Midpoint of Two Known Corners: Let's first explore the possible pairs to find the midpoints which can help us deduce the coordinates.
- Midpoint of \( A \) and \( C \): \[ M_{AC} = \left( \frac{1 + 9}{2}, \frac{7 + 7}{2} \right) = (5, 7) \]
- Midpoint of \( A \) and \( B \): \[ M_{AB} = \left( \frac{1 + 5}{2}, \frac{7 + 3}{2} \right) = (3, 5) \]
- Midpoint of \( B \) and \( C \): \[ M_{BC} = \left( \frac{5 + 9}{2}, \frac{3 + 7}{2} \right) = (7, 5) \]
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Identify Which Pairs are Diagonally Opposite: For a square, the midpoints that are relevant will help us find the remaining coordinates. If \( A \) and \( C \) are opposite corners, we need \( B \) and \( D(x, y) \) to satisfy the square's structure.
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Determine the Missing Corner: Since \( A \) and \( C \) might be opposites, we can find \( D \) by using vector addition or properties of the square in 2D. In a square, the diagonals bisect each other and are equal.
Using vectors:
- The vector from \( A \) to \( B \) is: \[ \vec{AB} = (5-1, 3-7) = (4, -4) \]
To find point \( D \), we can apply the vector from \( B \) in a way that mirrors the operation to get to \( D \).
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Perform the Vector Step: Move from \( C \) the same distance and in the same direction as \( \vec{AB} \) to find \( D \):
- Set \( D \) as: \[ D = C + \vec{AB} = (9 + 4, 7 - 4) = (13, 3) \]
However, we should validate possible combinations if \( A \) and \( B \) or any other pair are compared similarly to clarify corners.
Final Calculation of Fourth Corner: After evaluating configurations, if we consider various placements leading us to reevaluate: If \( A \) and \( B \) are opposite points, we derive:
Using coordinates of points, we can find other potential deductions leading to intersections of squares. Follow systematically any opposite two coordinate considerations to validate proper vertices based on 90-degree rotations possible with transformation angle properties yielding square-formational geometry.
The final coordinates of the fourth corner can be articulated as:
\[ D = (7, 9) \text{ or inversely mirrorable to a square placement reevaluation for explicit confirmation.} \]
Thus, total options based should provide direct relaying confirmation across standard placement checks yielding \( D \).