Question

Use the coordinates to compute the exact area of the rectangle.
Point A: (0, 0)
Point B: (4, -4)
Point C: (7, -1)
Point D: (3, 3)
A. 9 units^2
B. 20 units^2
C. 48 units^2
D. 24 units^2

Answer 1

To find the exact area of the rectangle defined by the given points, we first need to determine if these points represent a rectangle and then calculate its area.

The coordinates of the points are:
- Point A: (0, 0)
- Point B: (4, -4)
- Point C: (7, -1)
- Point D: (3, 3)

We will check if these points can form a rectangle. For a rectangle, opposite sides must be equal in length and the angles must be right angles.

1. **Calculating the lengths of the sides:**
- Side AB (between points A and B):
\[
\text{Length} = \sqrt{(4 - 0)^2 + (-4 - 0)^2} = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
\]

- Side BC (between points B and C):
\[
\text{Length} = \sqrt{(7 - 4)^2 + (-1 - (-4))^2} = \sqrt{(3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]

- Side CD (between points C and D):
\[
\text{Length} = \sqrt{(3 - 7)^2 + (3 - (-1))^2} = \sqrt{(-4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
\]

- Side DA (between points D and A):
\[
\text{Length} = \sqrt{(3 - 0)^2 + (3 - 0)^2} = \sqrt{(3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]

2. **Checking the lengths for opposite sides:**
- Side AB and CD have the same length: \( 4\sqrt{2} \)
- Side BC and DA have the same length: \( 3\sqrt{2} \)

3. **Determining the area of the rectangle:**
The area of a rectangle is calculated as the product of the lengths of two adjacent sides:
\[
\text{Area} = \text{Length}(AB) \times \text{Length}(BC) = (4\sqrt{2}) \times (3\sqrt{2}) = 12 \times 2 = 24
\]

Thus, the exact area of the rectangle is **24 square units**.

The correct answer is **D. 24 units^2**.