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

To determine the values of \( x \) and \( y \) such that quadrilateral ABCD is a parallelogram, we need to use the property of a parallelogram that states opposite sides are equal in length. Therefore, we can create equations based on the lengths of the sides provided.

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

For quadrilateral ABCD to be a parallelogram:
1. \( AB = CD \)
2. \( BC = AD \)

### Setting up the equations:

1. From \( AB = CD \):
\[
40x = 8x + 32
\]
Subtract \( 8x \) from both sides:
\[
32x = 32
\]
Dividing both sides by 32 gives:
\[
x = 1
\]

2. From \( BC = AD \):
\[
3y + 4 = 7y - 8
\]
Rearranging gives:
\[
4 + 8 = 7y - 3y
\]
\[
12 = 4y
\]
Dividing both sides by 4 gives:
\[
y = 3
\]

### Conclusion:
The values of \( x \) and \( y \) such that quadrilateral ABCD is a parallelogram are:
\[
x = 1, \quad y = 3
\]