Question

A (-3,-1)
B (1,5)
C (4,3)
D (0,-3)

Determine the coordinates of the rectangle, then compute the area.

a. 22 units^2
b. 11 units^2
c. 52 units^2
d. 26 units^2

Answer 1

To find the area of the rectangle formed by the points A(-3, -1), B(1, 5), C(4, 3), and D(0, -3), we first need to identify which points are the vertices of the rectangle.

Let's plot the points:

- A(-3, -1)
- B(1, 5)
- C(4, 3)
- D(0, -3)

To see which points form the rectangle, we need to check the distances between the points to determine the dimensions.

1. **Finding the distance between points:**

- Distance AB:
\[
\sqrt{(1 - (-3))^2 + (5 - (-1))^2} = \sqrt{(1 + 3)^2 + (5 + 1)^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52}
\]

- Distance AC:
\[
\sqrt{(4 - (-3))^2 + (3 - (-1))^2} = \sqrt{(4 + 3)^2 + (3 + 1)^2} = \sqrt{7^2 + 4^2} = \sqrt{49 + 16} = \sqrt{65}
\]

- Distance AD:
\[
\sqrt{(0 - (-3))^2 + (-3 - (-1))^2} = \sqrt{(0 + 3)^2 + (-3 + 1)^2} = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}
\]

- Distance BC:
\[
\sqrt{(4 - 1)^2 + (3 - 5)^2} = \sqrt{(3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}
\]

- Distance BD:
\[
\sqrt{(0 - 1)^2 + (-3 - 5)^2} = \sqrt{(-1)^2 + (-8)^2} = \sqrt{1 + 64} = \sqrt{65}
\]

- Distance CD:
\[
\sqrt{(0 - 4)^2 + (-3 - 3)^2} = \sqrt{(-4)^2 + (-6)^2} = \sqrt{16 + 36} = \sqrt{52}
\]

2. **Identifying rectangle vertices:**
The opposite sides must be equal in length:
- Sides AB and CD have the same length \(\sqrt{52}\).
- Sides AD and BC have the same length \(\sqrt{13}\).

Thus, the rectangle is defined by points A, B, D, and C.

3. **Finding the lengths of sides:**
- Length AB (or CD) = \(\sqrt{52} = 2\sqrt{13}\).
- Length AD (or BC) = \(\sqrt{13}\).

4. **Computing the area:**
The area \(Area\) of the rectangle is given by:
\[
Area = length \times width = (2\sqrt{13}) \times (\sqrt{13}) = 2 \cdot 13 = 26 \, \text{units}^2.
\]

Thus the area of the rectangle is **26 units²**.

The answer is:
d. 26 units²