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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=40
and y=16
x equals 40 and y equals 16

x=1
and y=4
x equals 1 and y equals 4
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1 answer

To determine for which values of \( x \) and \( y \) quadrilateral ABCD can be a parallelogram, we need to check the condition for a parallelogram: opposite sides must be equal in length.

Given:

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

For ABCD to be a parallelogram, the following must be true:

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

Let's set up the equations:

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

Now, let's solve each equation:

Solving the first equation:

\[ 40x = 8x + 32 \] Subtract \( 8x \) from both sides: \[ 32x = 32 \] Dividing both sides by 32 gives: \[ x = 1 \]

Solving the second equation:

\[ 4 + 3y = 9y - 20 \] Subtracting \( 3y \) from both sides: \[ 4 = 6y - 20 \] Adding 20 to both sides: \[ 24 = 6y \] Dividing both sides by 6 gives: \[ y = 4 \]

Conclusion

Using our results, we find: \[ x = 1 \quad \text{and} \quad y = 4 \]

Thus, the correct response is:

x equals 1 and y equals 4