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) x = y =
In order for ABCD to be a parallelogram, opposite sides must be equal.
We have:
AE = 3x - 6
DE = 6y - 12
EC = 2x + 2
EB = 2y + 24
In a parallelogram, AE = DC and DE = AB.
So, we can set up the following equations:
AE = DC ---> 3x - 6 = 6y - 12
DE = AB ---> 6y - 12 = 2y + 24
Simplifying these equations:
3x - 6 = 6y - 12
6y - 12 = 2y + 24
Subtracting 3x from both sides of the first equation and adding 2y to both sides of the second equation:
-3x + 6 = 6y - 6
2y - 2 = 2y + 24
Simplifying further:
-3x = 6y - 12
2y = 2y + 26
Subtracting 2y from both sides of the second equation:
-3x = 6
Now, we can solve for x:
x = -6 / -3
x = 2
Substituting the value of x into the first equation:
3x - 6 = 6y - 12
3(2) - 6 = 6y - 12
6 - 6 = 6y - 12
0 = 6y - 12
Adding 12 to both sides:
12 = 6y
Dividing both sides by 6:
2 = y
Therefore, x = 2 and y = 2 for ABCD to be a parallelogram.