Question

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?

Answer 1

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.