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.