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. (4 points)
For ABCD to be a parallelogram, opposite sides must be equal.
We have AE = DC and DE = BC.
So, 3x-6 = 2y+24 and 6y-12 = 2x+2.
Rearranging the first equation, we get 3x-2y = 30. (equation 1)
Rearranging the second equation, we get 6y-2x = 14. (equation 2)
Multiplying equation 1 by 3 and equation 2 by 2, we get:
9x-6y = 90. (equation 3)
12y-4x = 28. (equation 4)
Multiplying equation 4 by 9 and equation 3 by 4, we get:
36x-24y = 360. (equation 5)
36y-12x = 84. (equation 6)
Adding equation 5 and equation 6, we get:
-36x = 444
Dividing both sides by -36, we get:
x = -37/12.
Substituting x = -37/12 into equation 3, we get:
9(-37/12)-6y = 90.
-111/4 - 6y = 90.
-6y = 90 + 111/4.
-6y = 360 + 111/4.
-6y = 1470/4 + 111/4.
-6y = 1581/4.
y = -1581/24.
Therefore, x = -37/12 and y = -1581/24.