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Three vertices of a parallelogram are shown in the figure below. Give the coordinates of the fourth vertex.
Coordinates: (-3,8) (4,5) (2,-5)


Sagot :

Given:

The three vertices of a parallelogram are (-3,8), (4,5), (2,-5).

To find:

The fourth vertex of the parallelogram.

Solution:

Let the vertices of the parallelogram are A(-3,8), B(4,5), C(2,-5) and D(a,b).

We know that, diagonals of a parallelogram bisect each other. It means midpoints of both diagonals are same.

Midpoint formula:

[tex]Midpoint=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)[/tex]

Two diagonals of ABCD are AC and BD.

Midpoint of AC = Midpoint of BD

[tex]\left(\dfrac{-3+2}{2},\dfrac{8-5}{2}\right)=\left(\dfrac{4+a}{2},\dfrac{5+b}{2}\right)[/tex]

[tex]\left(\dfrac{-1}{2},\dfrac{3}{2}\right)=\left(\dfrac{4+a}{2},\dfrac{5+b}{2}\right)[/tex]

On comparing both sides, we get

[tex]\dfrac{4+a}{2}=\dfrac{-1}{2}[/tex]

[tex]4+a=-1[/tex]

[tex]a=-1-4[/tex]

[tex]a=-5[/tex]

And,

[tex]\dfrac{5+b}{2}=\dfrac{3}{2}[/tex]

[tex]5+b=3[/tex]

[tex]b=3-5[/tex]

[tex]b=-2[/tex]

Therefore, the coordinates of fourth vertex are (-5,-2).