91Ó°ÊÓ

The two-dimensional geometrical shape whose opposite sides are parallel to each other and equal in length is called a parallelogram. The interior opposite angles is also equal of a parallelogram.

To find the vertices through the equations, solve the equations first and then plot them on graph. The intersecting point between the lines will be the vertices of the parallelogram.

Consider the first equation.

$2x+y=-12$

To solve, first substitute xas zero to find the value of yand then substitute y as zero to find the value of x as shown.

When $x=0$,

$\begin{array}{c}2x+y=−12\\ 2\left(0\right)+y=−12\\ 0+y=−12\\ y=−12\end{array}$

When $y=0$,

$\begin{array}{c}2x+y=−12\\ 2x+0=−12\\ 2x=−12\\ x=−6\end{array}$

$\begin{array}{ccc}x& 0& −6\\ y& −12& 0\end{array}$

Repeat the same steps to solve all other equations.

Take second equation now.

$2x-y=-8$

To solve, first substitute xas zero to find the value of yand then substitute y as zero to find the value of x as shown.

When $x=0$,

$\begin{array}{c}2x−y=−8\\ 2\left(0\right)−y=−8\\ 0−y=−8\\ y=8\end{array}$

When$y=0$,

$\begin{array}{c}2x−y=−8\\ 2x−0=−8\\ 2x=−8\\ x=−4\end{array}$

The solution of the second equation is:

$\begin{array}{ccc}x& 0& −4\\ y& 8& 0\end{array}$

Let us take third equation now.

$2x-y-4=0$

To solve, first substitute xas zero to find the value of yand then substitute y as zero to find the value of x as shown.

When $x=0$,

$\begin{array}{c}2x−y−4=0\\ 2\left(0\right)−y−4=0\\ 0−y=4\\ y=−4\end{array}$
When $y=0$,

$\begin{array}{c}2x−y−4=0\\ 2x−0−4=0\\ 2x=4\\ x=2\end{array}$

The solution of the third equation is:

$\begin{array}{ccc}x& 0& 2\\ y& −4& 0\end{array}$

Now solve the fourth equation.

$4x+2y=24$

To solve, first substitute xas zero to find the value of yand then substitute y as zero to find the value of x as shown.

When $x=0$,

$\begin{array}{c}4x+2y=24\\ 4\left(0\right)+2y=24\\ 0+2y=24\\ y=12\end{array}$

When $y=0$,

$\begin{array}{c}4x+2y=24\\ 4x+2\left(0\right)=24\\ 4x=24\\ x=6\end{array}$

The solution of the fourth equation is:

$\begin{array}{ccc}x& 0& 6\\ y& 12& 0\end{array}$

The solutions of all the equations we got are:

Now plot all the coordinates on the graph and obtain the coordinates of the points where these lines intersect with each other.

The red line represents the equation $2x+y=-12$. The blue line represents the equation $2x-y=-8$. The green line represents the equation $2x-y-4=0$. And the purple line represents the equation $4x+2y=24$.

The points where these lines meet are the vertices of the parallelogram that is A$\left(-2 , -8\right)$, B$\left(-5,-2\right)$, C$\left(4,4\right)$, and D$\left(1,10\right)$represents the vertices of the parallelogram.

Hence, the vertices of the parallelogram are A$\left(-2 , -8\right)$, B$\left(-5,-2\right)$, C$\left(4,4\right)$, and D$\left(1,10\right)$.