To calculate the perimeter of the rectangle formed by the points A(-3, -1), B(-2, -4), C(4, -2), and D(3, 1), we first need to find the lengths of two adjacent sides of the rectangle (the base and the height).
Step 1: Calculate the lengths of the sides
We will use the distance formula, which is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Finding the length of AB:
- Point A: (-3, -1)
- Point B: (-2, -4)
Using the distance formula: \[ AB = \sqrt{((-2) - (-3))^2 + ((-4) - (-1))^2} = \sqrt{(-2 + 3)^2 + (-4 + 1)^2} \] \[ = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \text{ units} \]
Finding the length of AD:
- Point A: (-3, -1)
- Point D: (3, 1)
Using the distance formula: \[ AD = \sqrt{(3 - (-3))^2 + (1 - (-1))^2} = \sqrt{(3 + 3)^2 + (1 + 1)^2} \] \[ = \sqrt{(6)^2 + (2)^2} = \sqrt{36 + 4} = \sqrt{40} \text{ units} = \sqrt{4 \times 10} = 2\sqrt{10} \text{ units} \]
Step 2: Calculate the perimeter
The perimeter \(P\) of a rectangle is given by: \[ P = 2 \times (\text{base} + \text{height}) \] Taking the lengths we calculated:
- Base (AB) = \(\sqrt{10}\) units
- Height (AD) = \(2\sqrt{10}\) units
Thus, the perimeter is: \[ P = 2 \times (\sqrt{10} + 2\sqrt{10}) = 2 \times 3\sqrt{10} = 6\sqrt{10} \text{ units} \]
Conclusion
The calculated perimeter is: \[ P = 6\sqrt{10} \text{ units} \]
This corresponds to the option \(P = 6\sqrt{10}\) or about 19.0 units.