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)

Question

Responses

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

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

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

x = 1 and y = 4

Answer 1

To ensure that quadrilateral ABCD is a parallelogram, we need to set the opposite sides equal to each other:

1. Set \(AB\) equal to \(CD\): \[ 40y = 8y + 32 \]

2. Set \(BC\) equal to \(AD\): \[ 4 + 3x = 9x - 20 \]

Now, we can solve each equation:

Solving for \(y\):

From the equation \(40y = 8y + 32\):

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

Solving for \(x\):

From the equation \(4 + 3x = 9x - 20\):

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

Thus, the values of \(x\) and \(y\) that make quadrilateral ABCD a parallelogram are:

\[ x = 4 \quad \text{and} \quad y = 1 \]

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