Question

Use the coordinates to compute the exact area of the rectangle.

A(0,0), B(4,-4), C(7,-1), and D,(3,3)

9
24
48
20

Answer 1

To find the area of the rectangle formed by the points A(0,0), B(4,-4), C(7,-1), and D(3,3), we will first check whether these points indeed form a rectangle.

For four points to form a rectangle, the following conditions must hold:
1. The opposite sides must be equal in length.
2. The diagonals must also be equal in length.

Let's compute the distances between these points:

1. **Distance AB:**
\[
AB = \sqrt{(4-0)^2 + (-4-0)^2} = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
\]

2. **Distance BC:**
\[
BC = \sqrt{(7-4)^2 + (-1-(-4))^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]

3. **Distance CD:**
\[
CD = \sqrt{(3-7)^2 + (3-(-1))^2} = \sqrt{(-4)^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
\]

4. **Distance DA:**
\[
DA = \sqrt{(3-0)^2 + (3-0)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]

Now we check the diagonals:
1. **Distance AC:**
\[
AC = \sqrt{(7-0)^2 + (-1-0)^2} = \sqrt{7^2 + (-1)^2} = \sqrt{49 + 1} = \sqrt{50} = 5\sqrt{2}
\]

2. **Distance BD:**
\[
BD = \sqrt{(3-4)^2 + (3-(-4))^2} = \sqrt{(-1)^2 + (7)^2} = \sqrt{1 + 49} = \sqrt{50} = 5\sqrt{2}
\]

The distances reveal:
- Lengths of opposite sides:
- AB = CD = \( 4\sqrt{2} \)
- BC = DA = \( 3\sqrt{2} \)

- Lengths of diagonals:
- AC = BD = \( 5\sqrt{2} \)

Since we have verified that opposite sides are equal and diagonals are equal, points A, B, C, and D indeed form a rectangle.

Now we can calculate the area:
The area \( A \) of a rectangle is given by the formula:

\[
A = \text{length} \times \text{width}
\]

From the computed distances, the length is \( 4\sqrt{2} \) and the width is \( 3\sqrt{2} \).

\[
A = (4\sqrt{2}) \times (3\sqrt{2}) = 12 \times 2 = 24
\]

Thus, the exact area of the rectangle is \( \boxed{24} \).