ABCD is a square Points `E(4,3)` and `F(2,5)` lie on AB and CD, respectively,such that EF divides the square in two equal parts. If the coordinates of A are `(7,3)`,then the coordinates of other vertices can be

Options

A. `(7,2)`
B. `(7,5)`
C. `(-1,3)`
D. `(-1,5)`

Explanation

To solve the problem, we need to find the coordinates of the vertices B, C, and D of the square ABCD, given that A is at (7, 3) and points E(4, 3) and F(2, 5) lie on sides AB and CD, respectively, such that line EF divides the square into two equal parts.

### Step-by-Step Solution:

1. **Identify the coordinates of point A**:
- Given: A = (7, 3)

2. **Determine the coordinates of point E**:
- E lies on side AB, and its coordinates are given as E(4, 3). Since E has the same y-coordinate as A, this confirms that AB is horizontal.

3. **Find the midpoint O of line EF**:
- The coordinates of F are given as F(2, 5).
- Using the midpoint formula:
\[ O = \left( \frac{x_E + x_F}{2}, \frac{y_E + y_F}{2} \right) = \left( \frac{4 + 2}{2}, \frac{3 + 5}{2} \right) = \left( 3, 4 \right) \]

4. **Determine the coordinates of point B**:
- Since AB is horizontal, the y-coordinate of B will also be 3 (same as A).
- Let the x-coordinate of B be \( x_B \). Thus, B = (x_B, 3).
- The distance OA (from O to A) must equal the distance OB (from O to B) since O is the center of the square.
- Calculate the distance OA:
\[ OA^2 = (7 - 3)^2 + (3 - 4)^2 = 4 + 1 = 5 \]
- Calculate the distance OB:
\[ OB^2 = (x_B - 3)^2 + (3 - 4)^2 = (x_B - 3)^2 + 1 \]
- Setting OA² equal to OB²:
\[ 5 = (x_B - 3)^2 + 1 \]
\[ 4 = (x_B - 3)^2 \]
\[ x_B - 3 = \pm 2 \]
- Solving gives:
- \( x_B = 5 \) or \( x_B = 1 \)

5. **Determine the coordinates of point D**:
- Since AD is vertical, the x-coordinate of D will be the same as B.
- The y-coordinate of D will be the same as C (which we will find next).
- If B = (5, 3), then D = (5, y_D). If B = (1, 3), then D = (1, y_D).

6. **Determine the coordinates of point C**:
- Since CD is horizontal, the y-coordinate of C will be the same as D.
- If D = (5, y_D), then C = (7, y_D). If D = (1, y_D), then C = (7, y_D).
- Since EF divides the square into two equal parts, the y-coordinate of C will be 5 (the same as F).
- Thus, if D = (5, 5), then C = (7, 5). If D = (1, 5), then C = (7, 5).

7. **Final Coordinates**:
- The coordinates of the vertices are:
- A = (7, 3)
- B = (1, 3) or (5, 3)
- C = (7, 5)
- D = (1, 5) or (5, 5)

### Conclusion:
The coordinates of the vertices can be:
- A(7, 3), B(1, 3), C(7, 5), D(1, 5) or A(7, 3), B(5, 3), C(7, 5), D(5, 5).