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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=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 x equals 1 and y equals 4 x=40 and y=16

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For quadrilateral ABCD to be a parallelogram, the opposite sides must be equal. This means we need to set the following equations based on the side lengths:

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

Substituting the given expressions for side lengths:

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

Let's solve these equations one at a time.

For the first equation: \[ 40x = 8x + 32 \] Subtract \( 8x \) from both sides: \[ 40x - 8x = 32 \] \[ 32x = 32 \] Now divide by 32: \[ x = 1 \]

For the second equation: \[ 4 + 3y = 9y - 20 \] Add \( 20 \) to both sides: \[ 4 + 20 + 3y = 9y \] \[ 24 + 3y = 9y \] Subtract \( 3y \) from both sides: \[ 24 = 9y - 3y \] \[ 24 = 6y \] Now divide by 6: \[ y = 4 \]

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

Thus, Erik can be sure that his quadrilateral is a parallelogram for the values \( x = 1 \) and \( y = 4 \).

The correct response is: x equals 1 and y equals 4.