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)

Question

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