Question

Eric draws quadrilateral ABCD. If side AB=40x
, side BC=3y+4
,side CD=8x+32
and side AD=7y−8
, for what values of x
and y
can Erick be sure that his quadrilateral is a parallelogram?(1 point)
x=

y=

Answer 1

For quadrilateral ABCD to be a parallelogram, opposite sides must be equal. In this case, we have the following relationships:

1. \( AB = CD \)
2. \( BC = AD \)

Given the side lengths:
- \( AB = 40x \)
- \( BC = 3y + 4 \)
- \( CD = 8x + 32 \)
- \( AD = 7y - 8 \)

We can set up the equations based on the properties of parallelograms:

1. From \( AB = CD \):
\[
40x = 8x + 32
\]
To solve this, subtract \( 8x \) from both sides:
\[
40x - 8x = 32
\]
This simplifies to:
\[
32x = 32
\]
Dividing both sides by 32 gives:
\[
x = 1
\]

2. From \( BC = AD \):
\[
3y + 4 = 7y - 8
\]
Rearranging this equation, we subtract \( 3y \) from both sides:
\[
4 = 4y - 8
\]
Now, add 8 to both sides:
\[
12 = 4y
\]
Dividing both sides by 4 gives:
\[
y = 3
\]

Thus, for the quadrilateral to be a parallelogram, the values are:
\[
x = 1
\]
\[
y = 3
\]