Final answer:
To find the equation of the line that represents the other diagonal of the square, we need to determine the coordinates of the two endpoints of the diagonal that lies on the line 3x - y = 4. We can use the midpoint formula to find the coordinates of one endpoint of the diagonal, which is (2, 0). The equation of the line passing through the point (2, 0) with a slope of -1/3 is y = (-1/3)x + 2/3.
Step-by-step explanation:
To find the equation of the line that represents the other diagonal of the square, we need to determine the coordinates of the two endpoints of the diagonal that lies on the line 3x - y = 4. We already know that the center of the square is (1, -1), so let's find the coordinates of one endpoint of the diagonal. Since the diagonal passes through the center, we can use the midpoint formula to find the coordinates. The midpoint is the average of the x-coordinates and the average of the y-coordinates. The x-coordinate of the other endpoint is 1 + (4 - 1)/2 = 2, and the y-coordinate is -1 + (3)/2 = 0. Therefore, one endpoint of the diagonal is (2, 0).
Now, let's find the coordinates of the other endpoint by using the fact that the diagonal is perpendicular to the line 3x - y = 4. The slope of the line is 3, so the slope of the perpendicular diagonal is -1/3. The equation of the line passing through the point (2, 0) with a slope of -1/3 can be found using the point-slope form: y - 0 = (-1/3)(x - 2), which simplifies to y = (-1/3)x + 2/3. Therefore, the equation of the line that represents the other diagonal is y = (-1/3)x + 2/3.