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Ema drew quadrilateral ABCD. If side AB= 40y, side BC+ 4 + 3x, side CD= 8y+32, and side AD= 9x-20, for what values of x and y can Ema be sure that her quadrilateral is a parallelogram?

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To determine the values of \(x\) and \(y\) for which the quadrilateral ABCD can be a parallelogram, we need to apply one of the properties of parallelograms: the opposite sides must be equal.

According to Ema's definitions:

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

For quadrilateral ABCD to be a parallelogram, we must have:

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

Let's set up the equations based on these conditions:

  1. From \(AB = CD\): \[ 40y = 8y + 32 \]

    Subtract \(8y\) from both sides: \[ 40y - 8y = 32 \ 32y = 32 \]

    Divide both sides by \(32\): \[ y = 1 \]

  2. From \(BC = AD\): \[ 4 + 3x = 9x - 20 \]

    Rearranging the equation: \[ 4 + 20 = 9x - 3x \ 24 = 6x \]

    Divide both sides by \(6\): \[ x = 4 \]

Thus, the values of \(x\) and \(y\) that ensure quadrilateral ABCD is a parallelogram are: \[ x = 4 \quad \text{and} \quad y = 1 \]

In conclusion, \(x = 4\) and \(y = 1\) ensures that \(AB = CD\) and \(BC = AD\), confirming that ABCD is a parallelogram.