Solution:

To find the equations of the sides of the square, we need to determine the coordinates of the other two vertices. Since the opposite vertices of a square are equidistant from the center, we can use this property to find the missing vertices.

Finding the center of the square:
- Let's consider the coordinates of the given opposite vertices: (-2,3) and (8,5).
- The x-coordinate of the center can be found by taking the average of the x-coordinates of the given vertices: (8 + (-2)) / 2 = 6 / 2 = 3.
- Similarly, the y-coordinate of the center can be found by taking the average of the y-coordinates of the given vertices: (5 + 3) / 2 = 8 / 2 = 4.
- Therefore, the center of the square is (3,4).

Finding the coordinates of the other two vertices:
- Since the opposite vertices are equidistant from the center, we can determine the distance between the center and one of the given vertices.
- Let's consider the distance between the center (3,4) and the vertex (-2,3).
- The distance can be calculated using the distance formula: √[(x2 - x1)^2 + (y2 - y1)^2].
- Plugging in the values, we have √[(3 - (-2))^2 + (4 - 3)^2] = √[(3 + 2)^2 + 1^2] = √[5^2 + 1] = √[25 + 1] = √26.
- Therefore, the distance between the center and one of the given vertices is √26.

- Since the opposite vertices are equidistant from the center, the distance between the center and the other missing vertex will also be √26.
- We can use this distance to determine the coordinates of the missing vertices by considering all the possible combinations of adding or subtracting √26 from the center coordinates.
- The two possible combinations are: (3 ± √26, 4 ± √26).
- Therefore, the coordinates of the missing vertices are (3 + √26, 4 + √26) and (3 - √26, 4 - √26).

Finding the equations of the sides:
- Now that we have the coordinates of all four vertices of the square, we can determine the equations of the sides using the point-slope form of a line: y - y1 = m(x - x1).
- Let's consider the side formed by the vertices (-2,3) and (3 + √26, 4 + √26).
- We can find the slope of this side by using the formula: m = (y2 - y1) / (x2 - x1).
- Plugging in the values, we have m = (4 + √26 - 3) / (3 + √26 - (-2)) = (√26 + 1) / (5 + √26).
- Therefore, the equation of this side can be written as: y - 3 = [(√26 + 1) / (5 + √26)](x - (-2)).