Final answer:
The equation for the line containing the longer diagonal of the quadrilateral with vertices A(2, 2), B(-2, -2), C(1, -1), and D(6, 4) is found to be y = x after calculating the distances between A and B, and C and D, showing that the diagonal CD is longer.
Step-by-step explanation:
To find the equation of the line containing the longer diagonal of the quadrilateral with the given vertices A(2, 2), B(-2, -2), C(1, -1), and D(6, 4), we must first identify which pairs of points form the diagonals of the quadrilateral. By plotting these points or inspecting their coordinates, we can see that A and B are opposite vertices as well as C and D, so they will form the diagonals of the quadrilateral. The diagonal connecting A and B is easy to identify because their coordinates are mirror images of each other, indicating that the diagonal lies on a line where the x and y coordinates are always equal, which confirms option (a) as the equation. However, we must verify that this is the longer diagonal.
To verify, we calculate the distance between A and B using the distance formula: d = √((x_2 - x_1)^2 + (y_2 - y_1)^2), which results in d = √(((-2) - 2)^2 + ((-2) - 2)^2) = √((4)^2 + (-4)^2) = √(16 + 16) = √32.
The distance between C and D is: d = √((6 - 1)^2 + (4 - (-1))^2) = √((5)^2 + (5)^2) = √(25 + 25) = √50. Since √50 (≈ 7.07) is greater than √32 (≈ 5.66), we conclude the longer diagonal is CD. Therefore, the equation for the line containing the longer diagonal is the one with points C and D which indeed has a positive slope due to the increase in the y values as x values increase, confirming option (a) y = x is the correct equation for the longer diagonal of the quadrilateral.