Ema drew quadrilateral ABCD. If side AB = 40y, side BC = 4+3x, side CD = 8y+32, and side AAD = 9x-20, for what values of x and y can Ema be sure that her quadrilateral is a parallelogram?(1 point)

Question

Responses which answer is correct and why

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

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

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

x = 20 and y = 8

Answer 1

For a quadrilateral to be a parallelogram, opposite sides must be equal in length. In this case, we can set the lengths of opposite sides equal to each other and solve for \(x\) and \(y\).

The sides of the quadrilateral are given as follows:

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

For the quadrilateral to be a parallelogram, we need:

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

Let’s set up the equations:

1. From \( AB = CD \): \[ 40y = 8y + 32 \] Subtract \(8y\) from both sides: \[ 32y = 32 \] Divide by 32: \[ y = 1 \]

2. From \( BC = DA \): \[ 4 + 3x = 9x - 20 \] Rearranging gives: \[ 4 + 20 = 9x - 3x \] So, \[ 24 = 6x \] Dividing both sides by 6: \[ x = 4 \]

Now we have \( x = 4 \) and \( y = 1 \).

Thus, Ema can be sure that her quadrilateral is a parallelogram if: \[ \boxed{x = 4 \text{ and } y = 1} \]

This corresponds to the second response \( (x = 4 \text{ and } y = 1) \).