2. Points $A(-3,-1)$ and $B(1,-1)$ are adjacent vertices of a rectangle. Determine all possible pairs of the remaining vertices of the rectangle if its area is 24 square units.
2. Points $A(-3,-1)$ and $B(1,-1)$ are adjacent vertices of a rectangle. Determine all possible pairs of the remaining vertices of this rectangle if its area is 24 square units.

## Solution.

Points $A$ and $B$ determine a side of the rectangle with a length of 4 units.

1 POINT

If the area of the rectangle is 24 square units, we conclude that the side of the rectangle adjacent to side $\overline{A B}$ must have a length of 6 units.

The first possible pair of the remaining vertices is $C_{1}(1,5)$ and $D_{1}(-3,5)$.

The second possible pair of the remaining vertices is $C_{2}(1,-7)$ and $D_{1}(-3,-7)$.

![](https://cdn.mathpix.com/cropped/2024_05_30_1ae5e90785d026149547g-14.jpg?height=803&width=634&top_left_y=1780&top_left_x=240)

Note 1: The diagram is not necessary for obtaining all points, but the coordinates of the remaining vertices of the rectangle must be clearly highlighted.

Note 2: If the student has a diagram with the coordinates of the vertices of the rectangle clearly highlighted, as in the solution, but does not highlight the length of segment $\overline{A B}$ or the reasoning for the length of the other side of the rectangle, the total number of points should be reduced by 1 or 2 points.