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
To determine the values of x and y for which ABCD must be a parallelogram, we need to establish the relationships between the sides of the quadrilateral.
A parallelogram has opposite sides that are equal in length.
Given AE = 3x - 6 and EB = 2y + 24, we can equate these expressions to find a relationship between x and y:
AE = EB
3x - 6 = 2y + 24
Simplifying the equation, we have:
3x - 2y = 30 ---(1)
Similarly, let's equate DE and EC:
DE = EC
6y - 12 = 2x + 2
Simplifying the equation, we have:
2x - 6y = -14 ---(2)
We now have a system of two equations with two variables. Let's solve this system to find the values of x and y:
Equation (1) * 2 + Equation (2) * 3:
6x - 4y + 6x - 18y = 60 - 42
12x - 22y = 18
Divide both sides by 2:
6x - 11y = 9 ---(3)
Now, let's solve Equations (2) and (3). We will use the elimination method:
Multiply Equation (2) by 3:
6x - 18y = -42 ---(4)
Now, subtract Equation (4) from Equation (3):
(6x - 11y) - (6x - 18y) = 9 - (-42)
6y = 51
y = 51/6
y = 8.5
Substitute y = 8.5 back into Equation (1):
3x - 2(8.5) = 30
3x - 17 = 30
3x = 30 + 17
3x = 47
x = 47/3
x ≈ 15.67
Therefore, for ABCD to be a parallelogram, x ≈ 15.67 and y = 8.5.