Question

Use the coordinates to compute the exact area of the rectangle.
(1 point)
• 24 units2
O 48 units2
• 20 units2
• 9 units2
A(0,0)
B(4,-4)
C(7,-1)
D(3,3)

Answer 1

To compute the area of the rectangle formed by the given coordinates A(0,0), B(4,-4), C(7,-1), and D(3,3), we should confirm if these points indeed form a rectangle and then calculate the area accordingly.

The area of a rectangle can be calculated using the formula:

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

First, we will find the lengths of the sides using the distance formula which is given by:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

We will calculate the lengths of opposite sides to check if they are equal:

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

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

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

4. Calculate the distance AD:
\[
AD = \sqrt{(3 - 0)^2 + (3 - 0)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]

Since opposite sides are equal (AB = CD and BC = AD), we can conclude that these points do indeed form a rectangle.

Now we can calculate the area:

\[
\text{Area} = AB \times BC
\]

Substituting in the lengths:

\[
\text{Area} = (4\sqrt{2}) \times (3\sqrt{2}) = 4 \times 3 \times (\sqrt{2} \times \sqrt{2}) = 12 \times 2 = 24 \text{ units}^2
\]

Thus, the exact area of the rectangle is **24 units²**.