Final answer:
To find the fourth vertex of a parallelogram, the midpoint of the diagonals is used along with the properties that opposite sides are parallel and equal in length. The calculated coordinates for the fourth vertex are (-7, 4), which is not listed among the options provided. There might be an error in the given question or the options.
Step-by-step explanation:
To find the coordinates of the fourth vertex in a parallelogram, we can use the properties of a parallelogram which states that opposite sides are equal in length and parallel. We are given two opposite vertices of the parallelogram, (-2, 4) and (1, 0), and a third vertex, (-4, 0).
Let's denote the given vertices as A(-2, 4), B(1, 0), and C(-4, 0). Since D is the fourth vertex which we need to find, we can use the property that the diagonals of a parallelogram bisect each other to find the midpoint M of the diagonal AC. The midpoint formula is M = ((x1 + x2)/2, (y1 + y2)/2), where x1, y1, and x2, y2 are the coordinates of A and C respectively. So M = ((-2 - 4) / 2, (4 + 0) / 2) = (-3, 2).
Since M is also the midpoint of the diagonal BD, we can find the coordinates of D by using the fact that M will have the same distance from B as it does from D. Since we know the coordinates of B and M, we can calculate the coordinates of D:
Dx = 2 * Mx - Bx = 2 * (-3) - 1 = -6 - 1 = -7
Dy = 2 * My - By = 2 * 2 - 0 = 4 - 0 = 4
Therefore, the coordinates of the fourth vertex, D, are (-7, 4), which is not listed in the options provided. The coordinates must be recalculated if there's an error in the given options.