Question

The coordinates of three corners of a square are one, 7 5, 3 and 9, 7 what are the coordinates of the fourth corner of the square

Answer 1

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:

1. \( A(1, 7) \)
2. \( B(5, 3) \)
3. \( C(9, 7) \)

Let's denote the fourth corner as \( D(x, y) \).

### Steps to find D:

1. **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 \( C \):
\[
M_{BC} = \left( \frac{5 + 9}{2}, \frac{3 + 7}{2} \right) = (7, 5)
\]

2. **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.

3. **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 \( D \).

4. **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 \( D \).