Question

Isaiah draws a quadrilateral WXYZ. If side WX = x - 20, side XY = 6y + 4, side YZ = 4 - 3x, and WZ = 4y + 22, for what values of x and y can Isaiah be sure his quadrilateral is a parallelogram?

a. x = 1 and y = 4
b. x = 9 and y = 6
c. x = 16 and y = 9
d. x = 6 and y = 9

Answer 1

To determine the values of \(x\) and \(y\) for which the quadrilateral WXYZ is a parallelogram, we need to ensure that the opposite sides are equal, which gives us the following equations:

1. \(WX = YZ\)
2. \(XY = WZ\)

Substituting the given expressions for each side, we can set up the equations:

**Equation 1:**
\[
x - 20 = 4 - 3x
\]

**Equation 2:**
\[
6y + 4 = 4y + 22
\]

Now let's solve these equations one by one.

**Solving Equation 1:**
\[
x - 20 = 4 - 3x
\]
Adding \(3x\) to both sides:
\[
x + 3x - 20 = 4
\]
\[
4x - 20 = 4
\]
Adding 20 to both sides:
\[
4x = 24
\]
Dividing by 4:
\[
x = 6
\]

**Solving Equation 2:**
\[
6y + 4 = 4y + 22
\]
Subtracting \(4y\) from both sides:
\[
6y - 4y + 4 = 22
\]
\[
2y + 4 = 22
\]
Subtracting 4 from both sides:
\[
2y = 18
\]
Dividing by 2:
\[
y = 9
\]

Thus, we have \(x = 6\) and \(y = 9\).

Now let's check the options:

a. \(x = 1\) and \(y = 4\)
b. \(x = 9\) and \(y = 6\)
c. \(x = 16\) and \(y = 9\)
d. \(x = 6\) and \(y = 9\)

The correct answer is:

**d. \(x = 6\) and \(y = 9\)**.