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?

A. x= 20 and y = 8
B. x = 4 and y = 1
C. x = 1 and y = 4
D. x = 40 and y = 16

Answer 1

To determine the values of \(x\) and \(y\) for which quadrilateral ABCD is a parallelogram, we can use the property that in a parallelogram, opposite sides are equal in length. Therefore, we need to set up the following equations based on the given side lengths:

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

Substituting the given expressions for the sides, we get:

1. \(40x = 8x + 32\)
2. \(4 + 3y = 9y - 20\)

### Solve the first equation:

\[
40x = 8x + 32
\]

Subtract \(8x\) from both sides:

\[
40x - 8x = 32
\]
\[
32x = 32
\]

Now, divide by 32:

\[
x = 1
\]

### Solve the second equation:

\[
4 + 3y = 9y - 20
\]

First, move \(3y\) to the right side and \(-20\) to the left side:

\[
4 + 20 = 9y - 3y
\]
\[
24 = 6y
\]

Now divide by 6:

\[
y = 4
\]

### Conclusion

We have found that \(x = 1\) and \(y = 4\). Thus, the values that allow Erik to be sure that his quadrilateral is a parallelogram are:

**C. \(x = 1\) and \(y = 4\)**