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?(1 point)
Responses

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

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

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

x = 4 and y = 1

Answer 1

To check if quadrilateral ABCD is a parallelogram, we need to ensure that the opposite sides are equal. This gives us the following equations:

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

Substituting the given values into these equations, we have:

1. \( 40y = 8y + 32 \)
2. \( 4 + 3x = 9x - 20 \)

Now, let’s solve both equations.

### Solving Equation 1:
\[
40y = 8y + 32
\]
Subtract \( 8y \) from both sides:
\[
40y - 8y = 32
\]
This simplifies to:
\[
32y = 32
\]
Divide both sides by 32:
\[
y = 1
\]

### Solving Equation 2:
\[
4 + 3x = 9x - 20
\]
Subtract \( 3x \) from both sides:
\[
4 = 6x - 20
\]
Add 20 to both sides:
\[
24 = 6x
\]
Divide both sides by 6:
\[
x = 4
\]

Thus, we find:
\[
x = 4 \quad and \quad y = 1
\]

Now, let’s check back to see if this pair of values makes both pairs of opposite sides equal:

1. **For AB and CD**:
- \( AB = 40y = 40(1) = 40 \)
- \( CD = 8y + 32 = 8(1) + 32 = 8 + 32 = 40 \) (they are equal)

2. **For BC and AD**:
- \( BC = 4 + 3x = 4 + 3(4) = 4 + 12 = 16 \)
- \( AD = 9x - 20 = 9(4) - 20 = 36 - 20 = 16 \) (they are equal)

Since both pairs of opposite sides are equal, it confirms that quadrilateral ABCD is a parallelogram when \( x = 4 \) and \( y = 1 \).

Therefore, the correct answer is:
**x = 4 and y = 1**.