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