Question

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

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

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

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

x = 20 and y = 8

Answer 1

In a parallelogram, opposite sides are equal in length. Therefore, we can set up the following equations based on the sides of quadrilateral ABCD:

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

For sides given:

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

Using the first equation, we can set \( AB \) equal to \( CD \):

\[
40y = 8y + 32
\]

Now, solve for \( y \):

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

Next, we use the second equation. Set \( BC \) equal to \( AD \):

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

Now, solve for \( x \):

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

Thus, the values we found are \( x = 4 \) and \( y = 1 \).

So the correct response is:

**x = 4 and y = 1**