Question

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)
(x = 20) and (y = 8)
(x = 40) and (y = 16)
(x = 1) and (y = 4)
(x = 4) and (y = 1)

Explanation:

Step1: Use the property of parallelogram (opposite sides are equal)

For a parallelogram \(ABCD\), \(AB = CD\) and \(BC=AD\).
So, we have the equations:
\(40x=8x + 32\) (since \(AB = CD\))
\(4+3y=9y - 20\) (since \(BC = AD\))

Step2: Solve the first equation \(40x=8x + 32\)

Subtract \(8x\) from both sides:
\(40x-8x=8x + 32-8x\)
\(32x=32\)
Divide both sides by \(32\):
\(x=\frac{32}{32}=1\)

Step3: Solve the second equation \(4 + 3y=9y-20\)

Subtract \(3y\) from both sides:
\(4+3y-3y=9y-20 - 3y\)
\(4 = 6y-20\)
Add \(20\) to both sides:
\(4+20=6y-20 + 20\)
\(24=6y\)
Divide both sides by \(6\):
\(y=\frac{24}{6}=4\)

Answer:

\(x = 1\) and \(y = 4\) (the third option)