Question 10

Question

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.