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?(1 point)
Responses

x=40 and y=16
x equals 40 and y equals 16

x=1 and y=4
x equals 1 and y equals 4

x=4 and y=1
x equals 4 and y equals 1

x=20 and y=8

Answer 1

In a parallelogram, opposite sides must be equal. Therefore, we can set up the following equations based on the side lengths of quadrilateral ABCD:

1. **AB = CD**:
\[
40x = 8x + 32
\]

2. **BC = AD**:
\[
4 + 3y = 9y - 20
\]

Now let's solve these equations one by one.

### 1. Solve for \(x\):

Starting with:
\[
40x = 8x + 32
\]
Subtract \(8x\) from both sides:
\[
40x - 8x = 32
\]
\[
32x = 32
\]
Now divide both sides by 32:
\[
x = 1
\]

### 2. Solve for \(y\):

Next, we set up the second equation:
\[
4 + 3y = 9y - 20
\]
Start by adding \(20\) to both sides:
\[
4 + 20 + 3y = 9y
\]
Simplify the left side:
\[
24 + 3y = 9y
\]
Now, subtract \(3y\) from both sides:
\[
24 = 9y - 3y
\]
\[
24 = 6y
\]
Now divide both sides by 6:
\[
y = 4
\]

### Conclusion:
Thus, the values of \(x\) and \(y\) for which Erik can be sure that his quadrilateral is a parallelogram are:
\[
x = 1 \quad \text{and} \quad y = 4
\]

The correct choice is:
**x=1 and y=4** (or "x equals 1 and y equals 4").