To find the intersection point of the diagonals of a square, you can use the midpoint formula. This is because the diagonals of a square bisect each other at their midpoint. Given the points [tex]\( A = (3, 2) \)[/tex], [tex]\( B = (-1, 1) \)[/tex], [tex]\( C = (0, -3) \)[/tex], and [tex]\( D = (4, -2) \)[/tex], we first calculate the midpoints of the diagonals [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex].
### Step 1: Calculate the midpoint of diagonal [tex]\( AC \)[/tex]
The midpoint [tex]\( M_{AC} \)[/tex] of diagonal [tex]\( AC \)[/tex] is given by:
[tex]\[ M_{AC} = \left( \frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2} \right) \][/tex]
Substitute the coordinates of points [tex]\( A \)[/tex] and [tex]\( C \)[/tex]:
[tex]\[ M_{AC} = \left( \frac{3 + 0}{2}, \frac{2 + (-3)}{2} \right) \][/tex]
[tex]\[ M_{AC} = \left( \frac{3}{2}, \frac{-1}{2} \right) \][/tex]
[tex]\[ M_{AC} = (1.5, -0.5) \][/tex]
### Step 2: Calculate the midpoint of diagonal [tex]\( BD \)[/tex]
The midpoint [tex]\( M_{BD} \)[/tex] of diagonal [tex]\( BD \)[/tex] is given by:
[tex]\[ M_{BD} = \left( \frac{x_2 + x_4}{2}, \frac{y_2 + y_4}{2} \right) \][/tex]
Substitute the coordinates of points [tex]\( B \)[/tex] and [tex]\( D \)[/tex]:
[tex]\[ M_{BD} = \left( \frac{-1 + 4}{2}, \frac{1 + (-2)}{2} \right) \][/tex]
[tex]\[ M_{BD} = \left( \frac{3}{2}, \frac{-1}{2} \right) \][/tex]
[tex]\[ M_{BD} = (1.5, -0.5) \][/tex]
### Step 3: Intersection point of the diagonals
The intersection point of the diagonals of the square is where the two midpoints calculated above coincide. From our calculations, both [tex]\( M_{AC} \)[/tex] and [tex]\( M_{BD} \)[/tex] are the same:
[tex]\[ (1.5, -0.5) \][/tex]
Therefore, the coordinates of the point where the diagonals intersect are:
[tex]\[ \boxed{(1.5, -0.5)} \][/tex]
