Final answer:
The coordinates of the fourth vertex of the square are (1, 4), and the area of the square is 25 square units.
Step-by-step explanation:
To find the coordinates of the fourth vertex of the square, we can use the fact that a square has all sides of equal length and the sides are perpendicular to each other. We are given three vertices of the square: (1, -1), (6, -1), and (6, 4). The given points suggest that two sides of the square run parallel to the x-axis and two sides run parallel to the y-axis, forming right angles.
By examining the given points, we can see that the square extends 5 units along the x-axis (from x=1 to x=6) and also needs to extend 5 units along the y-axis. Given that one side is from (6, -1) to (6, 4), which is a move of 5 units upwards, the opposite side should also be a move of 5 units upwards from (1, -1). This puts the fourth vertex at (1, -1 + 5), which is (1, 4).
To calculate the area of the square, we can use the formula Area = side × side. Since one side of the square is 5 units long (the distance between (1, -1) and (6, -1)), the area is 5 × 5 = 25 square units.