Question

Erik draws quadrilateral ABCD. If side AB = 40x, side BC = 4 + 3y, side
CD = 8x + 32, and side AD = 9y - 20, for what values of x and y can Erik be sure that his
quadrilateral is a parallelogram?

Answer 1

To determine the values of \(x\) and \(y\) for which quadrilateral ABCD can be classified as a parallelogram, we need to use the property that in a parallelogram, opposite sides are equal in length. Hence, we will equate the lengths of opposite sides.

Given:
- Side \(AB = 40x\)
- Side \(BC = 4 + 3y\)
- Side \(CD = 8x + 32\)
- Side \(AD = 9y - 20\)

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

This gives us two equations:

1. **From the first pair of opposite sides**:
\[
40x = 8x + 32
\]
To solve this equation, first simplify:
\[
40x - 8x = 32
\]
\[
32x = 32
\]
Dividing both sides by 32:
\[
x = 1
\]

2. **From the second pair of opposite sides**:
\[
4 + 3y = 9y - 20
\]
Rearranging gives:
\[
4 + 20 = 9y - 3y
\]
\[
24 = 6y
\]
Dividing both sides by 6:
\[
y = 4
\]

### Conclusion
To ensure Erik's quadrilateral is a parallelogram, the values of \(x\) and \(y\) must be \(x = 1\) and \(y = 4\). Therefore, the values are:

\[
\boxed{(1, 4)}
\]