Question

eric draws quadrilateral ABCD. if side AB=40x, side BC=3y+4, side CD=8x+32and sideb AD=7y-8, for what values of x and y can erick be sure that his quadrilateral is a parallelogram?

Answer 1

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

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

According to the properties of a parallelogram, we can set the following equations based on opposite sides being equal:

1. \( AB = CD \)
\[
40x = 8x + 32
\]
2. \( BC = AD \)
\[
3y + 4 = 7y - 8
\]

Let's solve each equation.

### Solving for \( x \):

Starting with the first equation:
\[
40x = 8x + 32
\]
Subtract \( 8x \) from both sides:
\[
32x = 32
\]
Now divide both sides by 32:
\[
x = 1
\]

### Solving for \( y \):

Now let's solve the second equation:
\[
3y + 4 = 7y - 8
\]
Subtract \( 3y \) from both sides:
\[
4 = 4y - 8
\]
Add 8 to both sides:
\[
12 = 4y
\]
Now divide both sides by 4:
\[
y = 3
\]

### Conclusion:

The values of \( x \) and \( y \) that will make quadrilateral ABCD a parallelogram are:
\[
x = 1 \quad \text{and} \quad y = 3.
\]