Erik draws quadrilateral ABCD . If side AB=40x , side BC=4+3y , side CD=8x+32 , and side AD=9y−20 , for what values of x and y can Erik be sure that his quadrilateral is a parallelogram?(1 point)

Question

Responses

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

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

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

x=20 and y=8

Answer 1

In a parallelogram, opposite sides must be equal. Therefore, we can set up the following equations based on the side lengths of quadrilateral ABCD:

1. AB = CD: \[ 40x = 8x + 32 \]

2. BC = AD: \[ 4 + 3y = 9y - 20 \]

Now let's solve these equations one by one.

1. Solve for \(x\):

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

2. Solve for \(y\):

Next, we set up the second equation: \[ 4 + 3y = 9y - 20 \] Start by adding \(20\) to both sides: \[ 4 + 20 + 3y = 9y \] Simplify the left side: \[ 24 + 3y = 9y \] Now, subtract \(3y\) from both sides: \[ 24 = 9y - 3y \] \[ 24 = 6y \] Now divide both sides by 6: \[ y = 4 \]

Conclusion:

Thus, the values of \(x\) and \(y\) for which Erik can be sure that his quadrilateral is a parallelogram are: \[ x = 1 \quad \text{and} \quad y = 4 \]

The correct choice is: x=1 and y=4 (or "x equals 1 and y equals 4").