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**Hint:**We are given the three vertices of a parallelogram $ABCD$ are $A(3,-4),B(-1,-3)$ and $C(-6,2)$. By using midpoint $O$ find vertex $D$. After that, to find the area used, Area of parallelogram $ABCD$ $=$ Area of $\Delta ABC$ $+$ Area $\Delta ADC$.

**Complete step-by-step answer:**Now we are given the three vertices of a parallelogram $ABCD$ are $A(3,-4),B(-1,-3)$ and $C(-6,2)$.

$O$ is midpoint of $AC$$=(\dfrac{3-6}{2},\dfrac{-4+2}{2})$

Simplifying we get,

$O$ is midpoint of $AC$$=(\dfrac{-3}{2},-1)$ ……… (1)

In similar way,

$O$ is the midpoint of $BD$$=(\dfrac{-1+a}{2},\dfrac{-3+b}{2})$ …………. (2)

Now equating (1) and (2), we get,

$(\dfrac{-3}{2},-1)=(\dfrac{-1+a}{2},\dfrac{-3+b}{2})$

So, $-\dfrac{3}{2}=\dfrac{-1+a}{2}$

$a=-2$

Now, $\dfrac{-3+b}{2}=-1$

$b=1$

Therefore, $D(-2,1)$.

Now,

Area of parallelogram $ABCD$ $=$ Area of $\Delta ABC$ $+$ Area $\Delta ADC$

Area of parallelogram $ABCD$ $=$ $2$Area of $\Delta ABC$

Let us find Area of $\Delta ABC$ ,

Area of $\Delta ABC$ $=\dfrac{1}{2}[{{x}_{1}}({{y}_{2}}-{{y}_{3}})+{{x}_{2}}({{y}_{3}}-{{y}_{1}})+{{x}_{3}}({{y}_{1}}-{{y}_{2}})]$

Area of $\Delta ABC$ \[=\dfrac{1}{2}[3(1-2)-2(2+4)-6(-4-1)]\]

Simplifying we get,

Area of $\Delta ABC$ \[=\dfrac{15}{2}\]sq. units

So now,

Area of parallelogram $ABCD$ $=$ $2$Area of $\Delta ABC$$=2\times \dfrac{15}{2}=15$ sq. units

**Therefore, the area of parallelogram is $15$ square. units.**

**Additional information:**

A parallelogram is a two-dimensional geometrical shape, whose sides are parallel with each other. It is a polygon having four sides, where the pair of parallel sides are equal in length. Also, the interior opposite angles of a parallelogram are equal to each other. The area of parallelogram depends on the base and height of it. A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length, and the opposite angles are equal in measure. Also, the interior angles on the same side of the transversal are supplementary.

**Note:**A square and a rectangle are two shapes which have similar properties of a parallelogram. The opposite sides of a parallelogram are equal in length, and the opposite angles are equal in measure. The area of a parallelogram is the region bounded by the parallelogram in a given two-dimension space. To recall, a parallelogram is a special type of quadrilateral which has four sides, and the pair of opposite sides are parallel.

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