Question

Question 10
A)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?(1 point)
Responses

x = 40 and y = 16
x = 40 and y = 16

x = 20 and y = 8
x = 20 and y = 8

x = 4 and y = 1
x = 4 and y = 1

x = 1 and y = 4
x = 1 and y = 4

Answer 1

To determine the values of \(x\) and \(y\) for which quadrilateral \(ABCD\) is a parallelogram, we need the opposite sides to be equal. This leads us to set up the following equations based on the side lengths given:

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

Based on the problem, we have:

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

### Setting the equations:

**For the first pair of opposite sides:**

\[
40y = 8y + 32
\]

Subtract \(8y\) from both sides:

\[
40y - 8y = 32 \implies 32y = 32 \implies y = 1
\]

**For the second pair of opposite sides:**

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

Rearranging gives:

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

### Conclusion:

The values are \(x = 4\) and \(y = 1\). Therefore, the correct response is:

**x = 4 and y = 1**.