Chapter 6.2, Problem 41HP

To determine

The length of AB.

Answer to Problem 41HP

  $AB=7$

Explanation of Solution

Given:

ABCD is a parallelogram.

Perimeter of ABCD is 22.

Calculation:

The opposite sides in a parallelogram are congruent.

Therefore,

  $AB=DC,AD=BC$

So,

  $\begin{array}{l}2y+1=3-4w\left(\text{Equation 1}\right)\\ 3x-2=x-w+1\left(\text{Equation 2}\right)\end{array}$

Solving both the equations:

  $\begin{array}{l}2y+1=3-4w\\ 2y=-4w+3-1\\ 2y=-4w+2\\ y=-2w+1\end{array}$

  $\begin{array}{l}3x-2=x-w+1\\ 3x-x=-w+1+2\\ 2x=-w+3\\ x=-\frac{1}{2}w+\frac{3}{2}\end{array}$

As the perimeter is 22, so,

  $\begin{array}{l}AB+AD=\frac{1}{2}\left(22\right)\\ 2y+1+3x-2=11\\ 3-4w+3\left(-\frac{1}{2}w+\frac{3}{2}\right)-2=11\\ -\frac{11}{2}w+\frac{11}{2}=11\\ -\frac{11}{2}w=\frac{11}{2}\\ w=-1\end{array}$

The length of AB is:

  $\begin{array}{l}AB=CD\\ AB=3-4w\\ AB=3-4\left(-1\right)\\ AB=7\end{array}$