In parallelogram DEFG, DH = x + 1, HF=3y. GH3z4, and HE = 5y + 1. Find the values of x and y. The diagram is not drawn to scale.
(2 points)
To find the values of x and y, we can use the properties of a parallelogram:
1. In a parallelogram, opposite sides are equal in length. So we can set up the following equations:
DH = FG
x + 1 = 3z - 4
HE = FG
5y + 1 = 3z - 4
2. We also know that the diagonals of a parallelogram bisect each other. So DH = HF and HE = FG. Therefore:
x + 1 = 3y
5y + 1 = 3z - 4
Now we have a system of equations:
x + 1 = 3z - 4
x + 1 = 3y
5y + 1 = 3z - 4
We can solve this system of equations to find the values of x and y. Let's start by solving the second equation for x:
x = 3y - 1
Now we can substitute this expression for x into the first equation:
3y - 1 + 1 = 3z - 4
3y = 3z - 4
3y + 4 = 3z
Now we can substitute this expression for 3z into the third equation:
5y + 1 = 3(3y + 4) - 4
5y + 1 = 9y + 12 - 4
5y + 1 = 9y + 8
-4y = 7
y = -7/4
Now that we have found the value of y, we can substitute it back into the equation for x:
x = 3(-7/4) - 1
x = -21/4 - 4/4
x = -25/4
Therefore, the values of x and y are x = -25/4 and y = -7/4.