Question

Use the coordinates to compute the exact area of the rectangle

A=-3,-1 B=1,5 C=4,3 D=0,-3

Answer 1

To find the area of a rectangle given its vertices, we can first verify that the given points indeed form a rectangle and then use the coordinates to compute the area.

The vertices given are:
- A(-3, -1)
- B(1, 5)
- C(4, 3)
- D(0, -3)

### Step 1: Verify the structure of the rectangle
We need to check if the lengths of the sides formed by the points are perpendicular and equal in pairs.

1. Find the distances between the points:
- Distance AB:
\[
AB = \sqrt{(1 - (-3))^2 + (5 - (-1))^2} = \sqrt{(1 + 3)^2 + (5 + 1)^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}
\]
- Distance BC:
\[
BC = \sqrt{(4 - 1)^2 + (3 - 5)^2} = \sqrt{(3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}
\]
- Distance CD:
\[
CD = \sqrt{(4 - 0)^2 + (3 - (-3))^2} = \sqrt{(4)^2 + (3 + 3)^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}
\]
- Distance DA:
\[
DA = \sqrt{((-3) - 0)^2 + ((-1) - (-3))^2} = \sqrt{(-3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13}
\]

### Step 2: Determine if opposite sides are equal and perpendicular
From the above calculations, we can conclude:
- AB and CD are both \(2\sqrt{13}\)
- BC and DA are both \(\sqrt{13}\)

### Step 3: Calculate the area of the rectangle
To compute the area of the rectangle, we can use the length of two adjacent sides. Based on the distances:
- One pair of opposite sides is \(2\sqrt{13}\)
- The other pair of opposite sides is \(\sqrt{13}\)

Now we calculate the area:
\[
\text{Area} = \text{Length} \times \text{Width} = (2\sqrt{13}) \times (\sqrt{13}) = 2 \times 13 = 26
\]

### Conclusion
Thus, the exact area of the rectangle formed by the points A, B, C, and D is \( \boxed{26} \).