Question

$▱\mathrm{DECK}$is a parallelogram. Complete.

If $\mathrm{DE}=\mathrm{x}+\mathrm{y},\mathrm{EC}=12,\mathrm{CK}=2\mathrm{x}-\mathrm{y}$and $\mathrm{KD}=3\mathrm{x}-2\mathrm{y}$then and $\mathrm{x}=?$,$\mathrm{y}=?$, and the perimeter of$▱\mathrm{DECK}$

Short answer

The value of x and y is 6 and 3 respectively and the perimeter of a$▱DECK$ is 42.

Step-by-step solution

Step 1. Apply property of parallelogram.

Opposite sides of a parallelogram are congruent, that is, they are equal in length.

In a given figure, by the property of a parallelogram,

$\begin{array}{c}x+y=2x-y\\ x=2y;\\ 3x-2y=12\end{array}$

Step 2. Description of step.

Substitute 2y for x into $3x-2y=12$ to find the value of y.

$\begin{array}{c}3\left(2y\right)-2y=12\\ 6y-2y=12\\ 4y=12\\ y=3\end{array}$

Step 3. Description of step.

Substitute 3 for y into $x=2y$ to find the value of x.

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

Step 4. Description of step.

The perimeter of a parallelogram is given by

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

Therefore, the value of x and y is 6 and 3 respectively and the perimeter of a$▱DECK$ is 42.