To solve this problem, we need to:

1. Determine the coordinates of points $A$, $B$, $C$, and $D$ from the graph.
2. Use the distance formula to find the lengths of sides $AB$ and $BC$.
3. Calculate the area of the rectangle by multiplying the lengths of sides $AB$ and $BC$.

Step 1: Identify the coordinates

• Point $A$: $(-4, -3)$
• Point $B$: $(-2, 0)$
• Point $C$: $(2, -1)$
• Point $D$: $(0, -4)$

Step 2: Apply the distance formula

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• For $AB$, using points $A(-4, -3)$ and $B(-2, 0)$: $AB = \sqrt{(-2 - (-4))^2 + (0 - (-3))^2} = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13}$

• For $BC$, using points $B(-2, 0)$ and $C(2, -1)$: $BC = \sqrt{(2 - (-2))^2 + (-1 - 0)^2} = \sqrt{(4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17}$

Step 3: Calculate the area of the rectangle

The area of rectangle $ABCD$ is: $\text{Area} = AB \times BC = \sqrt{13} \times \sqrt{17} = \sqrt{13 \times 17} = \sqrt{221}$

So, the solutions are:

• The length of $AB$ is $\sqrt{13}$ units.
• The length of $BC$ is $\sqrt{17}$ units.
• The area of the rectangle is $\sqrt{221}$ square units.

Would you like further details or have any questions? Here are some related questions to extend this problem:

1. How would you find the diagonal length of the rectangle?
2. How can you verify that the shape is a rectangle using the slopes of the sides?
3. What are the slopes of the diagonals of the rectangle?
4. How would the area change if the shape were rotated on the coordinate plane?
5. How would you calculate the perimeter of the rectangle?

Tip: When dealing with distances on the coordinate plane, always ensure your points are accurately identified before applying the distance formula.