Final answer:
To calculate the area of a parallelogram given its vertices, you can either use the base and height method or apply the vector approach and use the distance formula to find the lengths of the diagonals. After calculating the lengths of the diagonals, the area is half the product of their lengths.
Step-by-step explanation:
To find the area of the parallelogram with vertices A(−2, 3), B(0, 6), C(4, 4), and D(2, 1), you can use the formula for the area of a parallelogram based on the coordinates of its vertices. The formula is:
Area = |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)| / 2
However, for a parallelogram, an easier method can be used by finding the base and the height. The base can be found by calculating the distance between two consecutive vertices, A and B for example, using the distance formula:
Base = √[(x2 - x1)^2 + (y2 - y1)^2]
If we choose AB as the base, the height can be determined as the perpendicular distance from vertex C or D to the line AB, which can be found using the concept of slopes or by using the area of triangles. However, as this particular problem doesn't require calculation of the base and height specifically, we can use vector methods or the distance formula directly for a different approach.
Using the distance formula: AC and BD are diagonals, the area of the parallelogram is half the product of the lengths of the diagonals.
Area = ½ |AC| × |BD|
After calculating the lengths of AC and BD using the distance formula, multiply them and divide by two to get the area of the parallelogram.