Use the coordinates to compute the approximate area of the rectangle

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

1 answer

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:

\[ \boxed{20} \]