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Eric draws quadrilateral ABCD. If side AB=40x

, side BC=3y+4
,side CD=8x+32
and side AD=7y−8
, for what values of x
and y
can Erick be sure that his quadrilateral is a parallelogram?(1 point)
x=

y=

1 answer

For quadrilateral ABCD to be a parallelogram, opposite sides must be equal. In this case, we have the following relationships:

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

Given the side lengths:

  • \( AB = 40x \)
  • \( BC = 3y + 4 \)
  • \( CD = 8x + 32 \)
  • \( AD = 7y - 8 \)

We can set up the equations based on the properties of parallelograms:

  1. From \( AB = CD \): \[ 40x = 8x + 32 \] To solve this, subtract \( 8x \) from both sides: \[ 40x - 8x = 32 \] This simplifies to: \[ 32x = 32 \] Dividing both sides by 32 gives: \[ x = 1 \]

  2. From \( BC = AD \): \[ 3y + 4 = 7y - 8 \] Rearranging this equation, we subtract \( 3y \) from both sides: \[ 4 = 4y - 8 \] Now, add 8 to both sides: \[ 12 = 4y \] Dividing both sides by 4 gives: \[ y = 3 \]

Thus, for the quadrilateral to be a parallelogram, the values are: \[ x = 1 \] \[ y = 3 \]