Final answer:
To find the fourth vertex (D) of the parallelogram, subtract vector BA from vertex C. Given A(-2, 1), B(2, 4), C(4, 2), the fourth vertex D is found to be (0, -1), completing the parallelogram.
Step-by-step explanation:
To determine the coordinates of the fourth vertex needed to form a parallelogram, you need to apply the concept of vector addition, as shown in the parallelogram rule. The given vertices (-2, 1), (2, 4), and (4, 2) suggest that we can treat the first two points as vectors, find their difference, and then add that difference to the third vertex to find the fourth vertex.
Lets say the vertices are A(-2, 1), B(2, 4), and C(4, 2) of the parallelogram. Then, AB = B - A = (2 + 2, 4 - 1) = (4, 3) and BC = C - B = (4 - 2, 2 - 4) = (2, -2). Thus, to find the coordinates of D, we use the vector AC = AB + BC. Since A is at (-2, 1), the coordinates of D can be found by adding AC to A: D = A + AC = A + AB + BC = (-2, 1) + (4, 3) + (2, -2) = (4, 2).
This indicates that the coordinates of the fourth vertex D are (4, 2), which we already have as vertex C. Therefore, there must be an error. Since a parallelogram has opposite sides that are equal, we should in fact subtract vector BA from C. That is D = C - BA = (4, 2) - (4, 3) = (0, -1). Hence, the correct coordinates for the fourth vertex D are (0, -1).