Final answer:
Quadrilateral ABCD is a parallelogram because its opposite sides are equal in length and parallel, confirmed by calculating distances and slopes.
Step-by-step explanation:
To decide whether the quadrilateral ABCD with A(-2, 1), B(1, 2), C(0, -1), and D(-3,-2) is a parallelogram, we can use the concept that in a parallelogram opposite sides are equal in length and parallel.
First, we compute the distances between opposite vertices. Using the distance formula √((x_2-x_1)^2+(y_2-y_1)^2), the lengths of AB and CD are:
- AB = √[(1 - (-2))^2 + (2 - 1)^2] = √[9 + 1] = √10
- CD = √[(-3 - 0)^2 + (-2 - (-1))^2] = √[9 + 1] = √10
And the lengths of BC and AD are:
- BC = √[(0 - 1)^2 + (-1 - 2)^2] = √[1 + 9] = √10
- AD = √[(-3 - (-2))^2 + (-2 - 1)^2] = √[1 + 9] = √10
Since the lengths of AB and CD are equal, as are the lengths of AD and BC, these pairs of opposite sides are equal in length. Therefore, ABCD could potentially be a parallelogram. However, we also need to check if the slopes of opposite sides are equal because parallelogram sides are parallel.
The slopes are calculated as follows:
- Slope of AB: (2 - 1) / (1 - (-2)) = 1 / 3
- Slope of CD: (-2 - (-1)) / (-3 - 0) = -1 / -3 = 1 / 3
- Slope of BC: (-1 - 2) / (0 - 1) = -3 / 1 = -3
- Slope of AD: (-2 - 1) / (-3 - (-2)) = -3 / -1 = 3
Opposite sides AB and CD have the same slope, as do opposite sides AD and BC. This confirms that opposite sides of quadrilateral ABCD are both equal in length and parallel, satisfying the criteria for a parallelogram.