Question

Ema drew quadrilateral ABCD. If side AB = 40y, side BC = 4+3x, side CD = 8y+32, and side AD = 9x-20, for what values of x and y can Ema be sure that her quadrilateral is a parallelogram?(1 point)
Responses

x = 20 and y = 8
x = 20 and y = 8

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

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

x = 1 and y = 4

Answer 1

To ensure that quadrilateral ABCD is a parallelogram, we need to set the opposite sides equal to each other:

1. Set \(AB\) equal to \(CD\):
\[
40y = 8y + 32
\]

2. Set \(BC\) equal to \(AD\):
\[
4 + 3x = 9x - 20
\]

Now, we can solve each equation:

### Solving for \(y\):

From the equation \(40y = 8y + 32\):

\[
40y - 8y = 32 \\
32y = 32 \\
y = 1
\]

### Solving for \(x\):

From the equation \(4 + 3x = 9x - 20\):

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

Thus, the values of \(x\) and \(y\) that make quadrilateral ABCD a parallelogram are:

\[
x = 4 \quad \text{and} \quad y = 1
\]

Therefore, the correct response is:
**x = 4 and y = 1**.