In parallelogram DEFG, DH =x+4, HF =3y, GH =3x-1, and HE =5y+3. Find the values of x and y. How long is DF?
H = midpoint of parallelogram
In this point the diagonals DF and GE of a parallelogram bisect each other.
By definition of parallelogram:
DH = HF
x + 4 = 3 y
and
GH = HE
3 x - 1 = 5 y + 3
Now you must solve system:
x + 4 = 3 y
3 x - 1 = 5 y + 3
_____________
Isolate x in equation:
x + 4 = 3 y
Subtract 4 to both sides
x = 3 y - 4
Put this value in equation:
3 x - 1 = 5 y + 3
3 ∙ ( 3 y - 4 ) - 1 = 5 y + 3
9 y - 12 - 1 = 5 y + 3
9 y - 13 = 5 y + 3
Subtract 5 y to both sides
4 y - 13 = 3
Add 13 to both sides
4 y = 16
y = 16 / 4
y = 4
x = 3 y - 4
x = 3 ∙ 4 - 4
x = 12 - 4
x = 8
The solutions are:
x = 8 , y = 4
Now:
DH = x + 4 , HF = 3 y
DH = 8 + 4 , HF = 3 ∙ 4
DH = 12 , HF = 12
DF = DH + HF
DF = 12 + 12
DF = 24
Where is H?
opposite sides have the same length, and the diagonals bisect each other.
That should get you started.
To find the values of x and y, we can use the properties of a parallelogram.
Since opposite sides of a parallelogram are equal in length, we have the following equations:
DH = GF (1)
HG = FE (2)
From equation (1), we can equate the lengths and substitute the given values:
x + 4 = 3y (3)
From equation (2), we can equate the lengths and substitute the given values:
3x - 1 = 5y + 3 (4)
Solving equations (3) and (4) simultaneously will give us the values of x and y.
Substituting equation (3) into equation (4), we have:
3( x + 4 ) - 1 = 5y + 3
Expanding and simplifying:
3x + 12 - 1 = 5y + 3
3x + 11 = 5y + 3
3x = 5y + 3 - 11
3x = 5y - 8
Divide both sides by 3 to solve for x:
x = (5y - 8) / 3 (5)
Substituting equation (5) into equation (3), we have:
(5y - 8) / 3 + 4 = 3y
Multiply both sides by 3 to eliminate the fraction:
5y - 8 + 12 = 9y
5y + 4 = 9y
4 = 9y - 5y
4 = 4y
y = 1
Substituting the value of y into equation (3), we can solve for x:
x + 4 = 3(1)
x + 4 = 3
x = 3 - 4
x = -1
So, the values of x and y are x = -1 and y = 1.
To find the length of DF, we can use the Pythagorean theorem. In the parallelogram DEFG, DF and GH are diagonal sides that form a right angle.
Using the Pythagorean theorem for triangle DFG, we have:
DF^2 = DG^2 + GF^2
Substituting the given values:
DF^2 = (3x - 1)^2 + (3y)^2
DF^2 = (3(-1) - 1)^2 + (3(1))^2
DF^2 = (-3 - 1)^2 + (3)^2
DF^2 = (-4)^2 + (3)^2
DF^2 = 16 + 9
DF^2 = 25
Taking the square root of both sides, we have:
DF = √25
DF = 5
Therefore, the length of DF is 5 units.
To find the values of x and y, we can set up a system of equations using the given information.
Since opposite sides of a parallelogram are congruent, we can set up two equations:
Equation 1: DH = GH
x + 4 = 3x - 1
Simplifying, we get:
2x = 5
x = 2.5
Equation 2: HF = HE
3y = 5y + 3
Subtracting 3y from both sides, we get:
0 = 2y + 3
Subtracting 3 from both sides, we get:
-3 = 2y
Dividing by 2, we get:
y = -1.5
So the values of x and y are x = 2.5 and y = -1.5.
To find the length of DF, we can use the fact that opposite sides of a parallelogram are congruent.
DF = GH = 3x - 1
Plugging in the value of x, we get:
DF = 3(2.5) - 1
DF = 7.5 - 1
DF = 6.5
Therefore, the length of DF is 6.5 units.