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.
find x and Y
To determine the values of x and y for which ABCD must be a parallelogram, we need to use the properties of a parallelogram.
In a parallelogram, opposite sides are equal in length. Therefore, AE must be equal to DC and DE must be equal to BC.
From the given information, we can set up the equation:
AE = DC
3x - 6 = 6y - 12
Simplifying the equation, we get:
3x - 6y = -6
Similarly, we can set up the equation for DE = BC:
DE = BC
6y - 12 = 2y + 24
Simplifying the equation, we get:
4y = 36
Dividing both sides of the equation by 4, we find:
y = 9
Now, we can substitute the value of y into either of the original equations to solve for x:
3x - 6 = 6y - 12
3x - 6 = 6(9) - 12
3x - 6 = 54 - 12
3x - 6 = 42
Adding 6 to both sides of the equation:
3x = 48
Dividing both sides of the equation by 3, we find:
x = 16
Therefore, the values of x and y for which ABCD must be a parallelogram are:
x = 16
y = 9