Final answer:
To find the other 3 points of a rectangle in 2-D space, given one point and the area, calculate side lengths using the area, then determine the coordinates of the remaining points based on these dimensions while accounting for the rectangle's properties.
Step-by-step explanation:
To find permutations of all the other 3 points of a rectangle in 2-D space, given one of its points and the area, you must first understand the properties of a rectangle. A rectangle has opposing sides that are equal in length. If one side is known or assumed, the other side can be calculated using the area.
For instance, if a rectangle has an area of 80 square units and one of its sides (the width) is 4 units, the other side (the length) must be 20 units since the area of a rectangle is calculated by multiplying the width and length (20 units * 4 units = 80 square units).
With one vertex of the rectangle known, label it as point A (x1, y1). Knowing the lengths of the sides, we can find the coordinates of the other three vertices. The second point, B, can lie either on the horizontal or vertical line passing through A, thus giving us two options. Once we choose a position for B, say horizontally on the same y-value as A but at a distance equal to the length of the rectangle, we can then place the third point, C, at the same x-value as B but with a y-value that differs from A by the width of the rectangle. Lastly, the fourth point, D, would be aligned vertically with A and horizontally with C.
Remember, the order in which we calculate the sides does not matter due to the commutative property of addition. This is analogous to the vector addition used in physics where the parallelogram rule is applied to determine resultant vectors resulting from the addition of multiple vectors.