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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=4 and y=1 x equals 4 and y equals 1 x=20 and y=8 x equals 20 and y equals 8 x=40 and y=16 x equals 40 and y equals 16 x=1 and y=4 x equals 1 and y equals 4 Skip to navigation

1 answer

To determine when quadrilateral ABCD is a parallelogram, we need to satisfy the condition that opposite sides are equal.

From the information given:

  1. Side AB = \( 40x \)
  2. Side BC = \( 4 + 3y \)
  3. Side CD = \( 8x + 32 \)
  4. Side AD = \( 9y - 20 \)

For ABCD to be a parallelogram, we require:

  • \( AB = CD \)
  • \( BC = AD \)

This gives us the following equations to solve:

  1. From \( AB = CD \): \[ 40x = 8x + 32 \] Subtract \( 8x \) from both sides: \[ 32 = 40x - 8x \] \[ 32 = 32x \] \[ x = 1 \]

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

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

Now let’s check the possible answers:

  • \( x = 4 \) and \( y = 1 \) is not correct.
  • \( x = 20 \) and \( y = 8 \) is not correct.
  • \( x = 40 \) and \( y = 16 \) is not correct.
  • \( x = 1 \) and \( y = 4 \) is indeed correct.

Final Answer: \( x = 1 \) and \( y = 4 \).