Question

erik draws quadrilateral ABCD. if side AB = 40x, side BC = 4 + 3y, side CD = 8x + 32m and side AD = 9y - 20, for what values of x and y can erik be sure that his quadrilateral is a parallelogram?
a. x = 20 and y = 8
b. x = 1 and y = 4
c. x = 4 and y = 1
d. x = 40 and y = 16

Answer 1

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

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

Substituting the lengths of the sides, we have:

1. From \( AB = CD \):
\[
40x = 8x + 32m
\]

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

Now let's solve these equations one by one.

**Solving the first equation:**

\[
40x = 8x + 32m
\]
Subtracting \(8x\) from both sides:
\[
32x = 32m
\]
Dividing both sides by 32:
\[
x = m
\]

**Solving the second equation:**

\[
4 + 3y = 9y - 20
\]
Rearranging gives:
\[
4 + 20 = 9y - 3y
\]
\[
24 = 6y
\]
Dividing both sides by 6:
\[
y = 4
\]

Now we have \(m = x\) and \(y = 4\). This means \(x\) can be any value as long as it is equal to \(m\).

To see if any given choices match this:

- For option **a.** \(x = 20\) and \(y = 8\): \(y\) does not match; discard.
- For option **b.** \(x = 1\) and \(y = 4\): valid, if \(m = 1\).
- For option **c.** \(x = 4\) and \(y = 1\): \(y\) does not match; discard.
- For option **d.** \(x = 40\) and \(y = 16\): \(y\) does not match; discard.

Based on the analysis, the only options that satisfy the condition are those where \(y = 4\). Therefore, the answer is:

**b. \(x = 1\) and \(y = 4\)**