Chapter 6.3, Problem 44PFA

To determine

To find the length of each side of the parallelogram.

Answer to Problem 44PFA

71 inches, 20 inches

Explanation of Solution

Given:

Perimeter of the parallelogram: 244 inches

Difference between the length of the sides: 20 inches

Calculation:

Let the length of the longest side of the parallelogram be $2x+3y$ .

The other side of the parallelogram is $4x-y$ .

Perimeter of the parallelogram is twice the sum of the two sides.

Also, the perimeter of the parallelogram is 244 inches.

  $\begin{array}{l}\Rightarrow 2((2x+3y)+(4x-y))=244\\ \Rightarrow 4x+6y+8x-2y=244\\ \Rightarrow 12x+4y=\mathrm{244.........................................}(1)\end{array}$

Also, the difference between the length of the parallelogram is 20 inches.

  $\begin{array}{l}\Rightarrow (2x+3y)-(4x-y)=20\\ \Rightarrow 2x+3y-4x+y=20\\ \Rightarrow -2x+4y=\mathrm{20.........................................}(2)\end{array}$

Subtract the equations (1) and (2).

  $\begin{array}{l}\Rightarrow (12x+4y)+(-2x+4y)=244-20\\ \Rightarrow 12x+2x+4y-4y=224\\ \Rightarrow 14x=224\\ \Rightarrow x=16\end{array}$

Substitute the value of x in equation (2)

  $\begin{array}{l}\Rightarrow -2\times 16+4y=20\\ \Rightarrow -32+4y=20\\ \Rightarrow 4y=20+32\\ \Rightarrow 4y=52\\ \Rightarrow y=13\end{array}$

Substitute the value of x and y in the given equations

  $\begin{array}{c}2x+3y=(2\times 16)+(3\times 13)\\ =32+39\\ =71\end{array}$

Also,

  $\begin{array}{c}-2x+4y=(-2\times 16)+(4\times 13)\\ =-32+52\\ =20\end{array}$

Conclusion:

Therefore, the sides of the parallelogram are 71 inches and 20 inches.