To find the lengths of sides AB and BC, as well as the area of rectangle ABCD, we first need to determine the coordinates of the points and the side lengths.
The coordinates of the points are:
- A = (0, 0)
- B = (4, -4)
- C = (7, -1)
- D = (3, 3)
Step 1: Calculate the length of side AB.
The length of side AB can be calculated using the distance formula between points A and B.
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] For points A(0, 0) and B(4, -4):
\[ AB = \sqrt{(4 - 0)^2 + (-4 - 0)^2} \] \[ = \sqrt{4^2 + (-4)^2} \] \[ = \sqrt{16 + 16} \] \[ = \sqrt{32} = 4\sqrt{2} \]
Step 2: Calculate the length of side BC.
Next, we calculate the length of side BC using points B and C.
For points B(4, -4) and C(7, -1):
\[ BC = \sqrt{(7 - 4)^2 + (-1 + 4)^2} \] \[ = \sqrt{3^2 + 3^2} \] \[ = \sqrt{9 + 9} \] \[ = \sqrt{18} = 3\sqrt{2} \]
Step 3: Calculate the area of rectangle ABCD.
The area of a rectangle is given by the product of the lengths of its sides.
\[ \text{Area} = AB \times BC = (4\sqrt{2}) \times (3\sqrt{2}) \] \[ = 12 \times 2 = 24 \]
Final Results:
- Length of side AB: \(4\sqrt{2}\)
- Length of side BC: \(3\sqrt{2}\)
- Area of rectangle ABCD: \(24\)
Therefore, the answer is:
- Length of side AB: \(4\sqrt{2}\)
- Length of side BC: \(3\sqrt{2}\)
- Area of rectangle ABCD: \(24\)