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)
Possible Answers:
x=4 and y=1 x equals 4 and y equals 1 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=20 and y=8

Answer 1

For quadrilateral ABCD to be a parallelogram, opposite sides must be equal in length. Therefore, we can set up the following equations based on the sides given:

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

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

Setting up the equations:

1. From \( AB = CD \):
\[
40x = 8x + 32
\]
Simplifying this gives:
\[
40x - 8x = 32 \implies 32x = 32 \implies x = 1
\]

2. From \( BC = AD \):
\[
4 + 3y = 9y - 20
\]
Rearranging gives:
\[
4 + 20 = 9y - 3y \implies 24 = 6y \implies y = 4
\]

Now we have \( x = 1 \) and \( y = 4 \).

Therefore, Erik can be sure that his quadrilateral ABCD is a parallelogram when \( x = 1 \) and \( y = 4 \).

Looking at the provided answer choices, the correct answer is:
- x equals 1 and y equals 4.