A square is formed by the points (4, 5), (12, 5), (12, -3), and (4, -3). The diagonals of the square intersect at which of the following points ?

Options

A. (8, 5)
B. (9, 6)
C. (8, 1)
D. (12, 1)

Explanation

To find the point where the diagonals of the square intersect, we will calculate the midpoint of one of the diagonals formed by the given points.

### Step-by-Step Solution:

1. **Identify the points of the square**: The vertices of the square are given as (4, 5), (12, 5), (12, -3), and (4, -3).

2. **Determine the diagonals**: The diagonals of the square can be formed by connecting opposite vertices. We can choose the points (4, 5) and (12, -3) as one diagonal, and (12, 5) and (4, -3) as the other diagonal.

3. **Calculate the midpoint of one diagonal**: We will calculate the midpoint of the diagonal formed by the points (4, 5) and (12, -3).
- Let (x1, y1) = (4, 5) and (x2, y2) = (12, -3).
- The formula for the midpoint (M) is given by:
\[ M = \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right) \]

4. **Substitute the values into the midpoint formula**:
- Calculate the x-coordinate of the midpoint:
\[ M_x = \frac{4 + 12}{2} = \frac{16}{2} = 8 \]
- Calculate the y-coordinate of the midpoint:
\[ M_y = \frac{5 + (-3)}{2} = \frac{5 - 3}{2} = \frac{2}{2} = 1 \]

5. **Final midpoint**: Thus, the midpoint of the diagonal (4, 5) and (12, -3) is:
\[ M = (8, 1) \]

6. **Conclusion**: The diagonals of the square intersect at the point (8, 1).