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 values of x and y for which ABCD must be a parallelogram, we must set the opposite sides of the quadrilateral equal to each other.
First, we can set AE equal to DC:
3x - 6 = 2y + 24.
Next, we can set DE equal to BC:
6y - 12 = 2x + 2.
To solve this system of equations, we can start by isolating x in the first equation:
3x = 2y + 30,
x = (2/3)y + 10.
Then we can substitute this expression for x into the second equation to solve for y:
6y - 12 = 2((2/3)y + 10) + 2,
6y - 12 = (4/3)y + 20 + 2,
6y - 12 = (4/3)y + 22.
Next, we can isolate y on one side of the equation by subtracting (4/3)y from both sides:
(18/3)y - (4/3)y = 22 + 12,
(14/3)y = 34.
Finally, we can solve for y by multiplying both sides by (3/14):
y = 34(3/14),
y = 51/7.
To find the value of x, we can substitute this value of y into the expression x = (2/3)y + 10:
x = (2/3)(51/7) + 10,
x = 102/21 + 210/21,
x = 312/21,
x = 104/7.
Therefore, when x = 104/7 and y = 51/7, ABCD must be a parallelogram.