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=4 and y=1 x equals 4 and y equals 1 x=20 and y=8 x equals 20 and y equals 8 x=1 and y=4

Answer 1

For quadrilateral ABCD to be a parallelogram, opposite sides must be equal. Thus, we need to set the sides equal in pairs:

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

Substituting the side lengths we have:

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

Now let's solve each equation.

**1st Equation:**

\[
40x = 8x + 32
\]

Subtract \( 8x \) from both sides:

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

Dividing both sides by 32:

\[
x = 1
\]

**2nd Equation:**

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

Rearranging gives:

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

Dividing both sides by 6:

\[
y = 4
\]

So we have found that \( x = 1 \) and \( y = 4 \).

Now, we check the given options to confirm:

- \( \text{x = 1 and y = 4} \) is an option.

Thus, the correct values for \( x \) and \( y \) that ensure the quadrilateral is a parallelogram are:

**x = 1 and y = 4**.