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?
(x = 2) and (y = 8)
(x = 1) and (y = 4)
(x = 4) and (y = 1)
(x = 8) and (y = 16)

Explanation:

Step1: Use property of parallelogram

In a parallelogram, opposite - sides are equal. So, \(AB = CD\) and \(BC = AD\).
Set up the first equation from \(AB = CD\):
\(40x=8x + 32\)

Step2: Solve the first equation for \(x\)

Subtract \(8x\) from both sides: \(40x-8x=8x + 32-8x\), which gives \(32x=32\). Then divide both sides by 32: \(x = 1\).

Step3: Set up the second equation from \(BC = AD\)

\(4 + 3y=9y-20\)

Step4: Solve the second equation for \(y\)

First, add 20 to both sides: \(4 + 3y+20=9y-20 + 20\), which simplifies to \(24+3y=9y\). Then subtract \(3y\) from both sides: \(24+3y-3y=9y-3y\), getting \(24 = 6y\). Divide both sides by 6: \(y = 4\).

Answer:

\(x = 1\) and \(y = 4\)