Question

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?

Answer 1

Where is H?
opposite sides have the same length, and the diagonals bisect each other.
That should get you started.

Answer 2

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