Question

figure abcd is a parallelogram.
what is the perimeter of abcd?
14 units
38 units
44 units
49 units

Explanation:

Step1: Recall properties of parallelogram

In a parallelogram, opposite sides are equal. So, \(AB = CD\) and \(AD = BC\).
Given \(AB = 4y - 2\), \(CD = 2y + 6\), \(AD = 3x - 1\), \(BC = 2x + 2\).

First, solve for \(y\) using \(AB = CD\):
\(4y - 2 = 2y + 6\)
Subtract \(2y\) from both sides: \(4y - 2y - 2 = 6\)
Simplify: \(2y - 2 = 6\)
Add 2 to both sides: \(2y = 6 + 2 = 8\)
Divide by 2: \(y = \frac{8}{2} = 4\)

Now find length of \(AB\) (and \(CD\)): \(AB = 4(4) - 2 = 16 - 2 = 14\)

Step2: Solve for \(x\) using \(AD = BC\)

\(3x - 1 = 2x + 2\)
Subtract \(2x\) from both sides: \(3x - 2x - 1 = 2\)
Simplify: \(x - 1 = 2\)
Add 1 to both sides: \(x = 2 + 1 = 3\)

Find length of \(AD\) (and \(BC\)): \(AD = 3(3) - 1 = 9 - 1 = 8\)

Step3: Calculate perimeter of parallelogram

Perimeter of parallelogram \(= 2(AB + AD)\)
Substitute \(AB = 14\) and \(AD = 8\):
Perimeter \(= 2(14 + 8) = 2(22) = 44\)

Answer:

44 units