Question

A (-3,-2)
B (4,-2)
C (4,1)
D (-3,1)

Determine the coordinates of rectangle ABCD. Use the coordinates to compute the area of the rectangle.

___ units

Answer 1

To determine the coordinates and area of rectangle ABCD, let's first visualize the vertices based on the provided coordinates:

- Point A: (-3, -2)
- Point B: (4, -2)
- Point C: (4, 1)
- Point D: (-3, 1)

Next, let's check if these points indeed form a rectangle. A rectangle must have opposite sides equal in length and all angles equal to 90 degrees.

### Vertices and Lengths:

1. **AB**: The distance between points A and B can be calculated as:
\[
AB = |x_B - x_A| = |4 - (-3)| = |4 + 3| = 7
\]

2. **BC**: The distance between points B and C is:
\[
BC = |y_C - y_B| = |1 - (-2)| = |1 + 2| = 3
\]

3. **CD**: The distance between points C and D is:
\[
CD = |x_D - x_C| = |-3 - 4| = |-7| = 7
\]

4. **DA**: The distance between points D and A is:
\[
DA = |y_A - y_D| = |-2 - 1| = |-3| = 3
\]

### Checking Rectangle Properties:
- We see that opposite sides are equal: \( AB = CD = 7 \) and \( BC = DA = 3 \).
- Since there are two pairs of equal opposite sides, and the angles between these sides are 90 degrees (given the coordinate system), it confirms that ABCD forms a rectangle.

### Area Calculation:
The area \( A \) of a rectangle can be computed using the formula:
\[
\text{Area} = \text{length} \times \text{width}
\]

From our calculations:
- Length (AB or CD) = 7 units
- Width (BC or DA) = 3 units

Thus, the area is:
\[
\text{Area} = 7 \times 3 = 21 \text{ square units}
\]

Therefore, the area of rectangle ABCD is \(\boxed{21}\) units square.