To find the coordinates of the fourth vertex opposite to (3,1-7), we can use the fact that the opposite sides of a parallelogram are equal in length and parallel to each other.
Let's label the given vertices as A (3,7), B (5,-7), and C (-2,5). We need to find the coordinates of the fourth vertex, which we'll label as D.
First, we find the length and direction of the diagonal AC. The length of AC can be calculated using the distance formula:
AC = √[(x2 - x1)^2 + (y2 - y1)^2]
AC = √[(-2 - 3)^2 + (5 - 7)^2]
AC = √[(-5)^2 + (-2)^2]
AC = √[25 + 4]
AC = √29
Since opposite sides of a parallelogram are equal in length, the length of BD is also √29.
Next, we find the direction of the diagonal AC. The direction can be determined by finding the difference in x-coordinates and y-coordinates:
Δx = x2 - x1 = -2 - 3 = -5
Δy = y2 - y1 = 5 - 7 = -2
To find the coordinates of D, we add the differences to the coordinates of point B:
D(x, y) = B(x, y) + (Δx, Δy)
D(x, y) = (5, -7) + (-5, -2)
D(x, y) = (5 - 5, -7 - 2)
D(x, y) = (0, -9)
Therefore, the coordinates of the fourth vertex D, opposite to (3,1-7), are (0, -9).