Question

Use the coordinates to compute the approximate area of the rectangle
A(-3,-1)
B(-2,-4)
C(4,-2)
D(3,1)

Answer 1

To find the area of the rectangle defined by the points \( A(-3, -1) \), \( B(-2, -4) \), \( C(4, -2) \), and \( D(3, 1) \), we first need to determine the lengths of the sides of the rectangle.

1. **Identifying vertices vs. rectangle configuration**:
The points need to be labeled properly as the corners of the rectangle. By visually plotting or analyzing, we can rearrange these points:

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

2. **Finding the lengths of the sides**:
We will calculate the distances between the points to find the side lengths. The rectangle's area is given by the product of its length and width.

- Distance between points \( A \) and \( B \):
\[
d_{AB} = \sqrt{(-2 - (-3))^2 + (-4 - (-1))^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}
\]

- Distance between points \( A \) and \( D \):
\[
d_{AD} = \sqrt{(3 - (-3))^2 + (1 - (-1))^2} = \sqrt{(6)^2 + (2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10}
\]

3. **Area Calculation**:
Assuming \( d_{AB} \) represents one dimension (width), and \( d_{AD} \) represents the other dimension (length):
\[
\text{Area} = \text{Width} \times \text{Length} = d_{AB} \times d_{AD} = \sqrt{10} \times 2\sqrt{10} = 2 \times 10 = 20
\]

Hence, the approximate area of the rectangle formed by the points is:

\[
\boxed{20}
\]