Question

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

Answer 1

To determine if quadrilateral ABCD is a parallelogram, we need to ensure that both pairs of opposite sides are equal.

The sides are given as follows:

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

We set up the equations for the pairs of opposite sides:

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

Substituting in the side lengths, we have:

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

Starting with the first equation:

### Solving for \( y \):
\[
40y = 40 \implies y = 1
\]

Next, we solve the second equation:

### Solving for \( x \):

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

Rearranging gives:

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

### Checking the Values:

With \( y = 1 \) and \( x = 4 \):
- \( AB = 40y = 40(1) = 40 \)
- \( CD = 40 \)
- \( BC = 4 + 3(4) = 4 + 12 = 16 \)
- \( AD = 9(4) - 20 = 36 - 20 = 16 \)

Both pairs of opposite sides are equal:
- \( AB = CD \) results in \( 40 = 40 \)
- \( BC = AD \) results in \( 16 = 16 \)

### Conclusion:
The values that ensure quadrilateral ABCD is a parallelogram are \( x = 4 \) and \( y = 1 \).