Question

figure abcd is a parallelogram. what are the lengths of line segments ab and bc? ab = 4; bc = 16 ab = 4; bc = 8 ab = 10; bc = 20 ab = 10; bc = 28

Explanation:

Step1: Use the property of parallelogram (opposite sides are equal) for sides \(AD\) and \(BC\)

In parallelogram \(ABCD\), \(AD = BC\).
Given \(AD=2x - 4\) and \(BC=x + 12\), so \(2x-4=x + 12\).
Subtract \(x\) from both sides: \(2x-x-4=x-x + 12\), which gives \(x-4=12\).
Add \(4\) to both sides: \(x=12 + 4=16\).

Step2: Calculate the length of \(BC\)

Substitute \(x = 16\) into the expression for \(BC\): \(BC=x + 12=16+12 = 28\).

Step3: Use the property of parallelogram (opposite sides are equal) for sides \(AB\) and \(DC\)

In parallelogram \(ABCD\), \(AB = DC\).
Given \(AB=3y-2\) and \(DC=y + 6\), so \(3y-2=y + 6\).
Subtract \(y\) from both sides: \(3y-y-2=y-y + 6\), which gives \(2y-2=6\).
Add \(2\) to both sides: \(2y=6 + 2=8\).
Divide both sides by \(2\): \(y=\frac{8}{2}=4\).

Step4: Calculate the length of \(AB\)

Substitute \(y = 4\) into the expression for \(AB\): \(AB=3y-2=3\times4-2=12 - 2=10\).

Answer:

AB = 10; BC = 28 (the fourth option)