Chapter 5.1, Problem 26WE

To determine

To fill: in the blanks withthe values of x and y as well as the perimeter for a given parallelogram.

Answer to Problem 26WE

x = $6$

y = $3$

Perimeter of $▱DECK$$=42$

Explanation of Solution

Given information:

A diagram representing a parallelogram with measure of sides being given.

Formula used:

(1) By theorem 5-1, the opposite sides of a parallelogram are congruent to each other, which imply their measures being equal.

(2) The perimeter of a parallelogram with sides a and b is given by,

  $P=2(a+b)$

Calculation:

Consider the figure given below.

  

Based on the Theorem 5-1, it can be said that the opposite sides of the given parallelogram has the same value. This can be expressed as

  $\overline{DE}\cong \overline{CK}$

  $\overline{CE}\cong \overline{KD}$

Substituting the value of the sides based on the figure, the above two equations become,

  $\begin{array}{c}x+y=2x-y\\ x=2y\end{array}$

  $x=2y$ ……. (1)

  $3x-2y=12$

Substituting the (1) in the above equation,

  $\begin{array}{c}3x-x=12\\ 2x=12\\ x=\frac{12}{2}\\ =6\end{array}$

Now, using (1), the value of y can be calculated as,

  $\begin{array}{c}y=\frac{x}{2}\\ =\frac{6}{2}\\ =3\end{array}$

Now, perimeter of the parallelogram is given by

  $\begin{array}{c}P=2(\overline{DE}+\overline{EC})\\ =2(x+y+12)\\ =2(6+3+12)\\ =42\end{array}$