If AE=3x-6, DE = 6y-12, EC = 2x+2 and EB = 2y+24, find the values
of x and y for which ABCD must be a parallelogram. The drawing is not to scale.
To determine the conditions for ABCD to be a parallelogram, we must consider the opposite sides of ABCD.
If ABCD is a parallelogram, then AB must be parallel to CD, and thus the slopes of AB and CD must be equal.
The slope of AB is equal to the change in y divided by the change in x, so the slope of AB is:
(EB - EA) / (BC - BA)
Plugging in the given values:
(2y + 24 - (3x - 6)) / (EC - AE)
Simplifying:
(2y + 30 - 3x) / (2x + 2 - 3x + 6)
Expanding:
(2y - 3x + 30) / (2 - x)
This slope must be equal to the slope of CD, which can be found similarly as:
(EF - ED) / (DC - DE)
Plugging in the given values:
((3x + 6) - (6y - 12)) / (CD - DE)
Simplifying:
(3x - 6y + 18) / (DC - 6y + 12)
This slope must be equal to the slope of AB:
(2y - 3x + 30) / (2 - x) = (3x - 6y + 18) / (DC - 6y + 12)
To simplify this equation, we can cross multiply:
(2y - 3x + 30)(DC - 6y + 12) = (3x - 6y + 18)(2 - x)
Expanding:
2yDC - 12y - 3xDC + 18x + 30DC - 180 = 6x - 12y + 36 - 3xDC + 6yDC - 36x
Rearranging:
(2yDC - 12y + 30DC + 18x) + (12y - 6yDC + 36x - 3xDC) = (6x - 3xDC - 180 + 6yDC - 36 - 12y + 30DC)
Simplifying:
2yDC + 30DC + 18x = 6x - 3xDC + 6yDC - 12y - 30
Combining like terms:
(2yDC - 3xDC + 6yDC) + (30DC - 18x - 6x + 12y) = -12y - 30 + 30DC
Simplifying:
5yDC - 6xDC - 24x + 12y = -12y
Moving all terms to one side:
5yDC - 6xDC - 24x + 12y + 12y + 12 = 0
Simplifying:
5yDC - 6xDC - 24x + 24y + 12 = 0
Now we have an equation relating x and y that must be true for ABCD to be a parallelogram. We can solve this equation to find the values for x and y that satisfy this equation.