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)Responsesx=40 and y=16  x equals 40  and  y equals 16x=4 and y=1    x equals 4  and  y equals 1x=20 and y=8      x equals 20  and  y equals 8  x=1 and y=4x equals 1  and  y equals 4

Answer 1

For a quadrilateral to be a parallelogram, opposite sides must be equal in length. In quadrilateral ABCD, we have the sides:

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

From the properties of a parallelogram, we can establish the following equations based on the opposite sides being equal:

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

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

Now we can solve these equations step by step.

### Solve for x:
1. From \( 40x = 8x + 32 \):
\[
40x - 8x = 32 \\
32x = 32 \\
x = 1
\]

### Solve for y:
2. From \( 4 + 3y = 9y - 20 \):
\[
4 + 20 = 9y - 3y \\
24 = 6y \\
y = 4
\]

Thus, we find \( x = 1 \) and \( y = 4 \).

The correct response is:
**x = 1 and y = 4.**