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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?

A. x= 20 and y = 8
B. x = 4 and y = 1
C. x = 1 and y = 4
D. x = 40 and y = 16

1 answer

To determine the values of \(x\) and \(y\) for which quadrilateral ABCD is a parallelogram, we can use the property that in a parallelogram, opposite sides are equal in length. Therefore, we need to set up the following equations based on the given side lengths:

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

Substituting the given expressions for the sides, we get:

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

Solve the first equation:

\[ 40x = 8x + 32 \]

Subtract \(8x\) from both sides:

\[ 40x - 8x = 32 \] \[ 32x = 32 \]

Now, divide by 32:

\[ x = 1 \]

Solve the second equation:

\[ 4 + 3y = 9y - 20 \]

First, move \(3y\) to the right side and \(-20\) to the left side:

\[ 4 + 20 = 9y - 3y \] \[ 24 = 6y \]

Now divide by 6:

\[ y = 4 \]

Conclusion

We have found that \(x = 1\) and \(y = 4\). Thus, the values that allow Erik to be sure that his quadrilateral is a parallelogram are:

C. \(x = 1\) and \(y = 4\)