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 APARRALELOGRAM?

Answer 1

To determine the values of \(X\) and \(Y\) such that quadrilateral \(ABCD\) is a parallelogram, we need to use the property that in a parallelogram, opposite sides are equal in length.

Given the sides:
- \(AB = 40X\)
- \(BC = 4 + 3Y\)
- \(CD = 8X + 32\)
- \(AD = 9Y - 20\)

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

### First equation (for opposite sides):
\[
40X = 8X + 32
\]

### Solve for \(X\):
\[
40X - 8X = 32
\]
\[
32X = 32
\]
\[
X = 1
\]

### Second equation (for opposite sides):
\[
4 + 3Y = 9Y - 20
\]

### Solve for \(Y\):
\[
4 + 20 = 9Y - 3Y
\]
\[
24 = 6Y
\]
\[
Y = 4
\]

### Conclusion:
The values of \(X\) and \(Y\) that ensure quadrilateral \(ABCD\) is a parallelogram are:
\[
X = 1 \quad \text{and} \quad Y = 4
\]