Question

Show that the points (-2,-1), (1,0), (4,3) and (1,2) taken in order are
the vertices of a parallelogram.​

Answer 1

Given :-

Let us assume the vertices as A, B, C, D such that

• A (-2,-1)

• B (1,0)

• C (4,3)

and

• D (1,2)

To Prove :-

• ABCD is a parallelogram.

Concept Used :-

We know,

In parallelogram, diagonals bisect each other.

So in order to prove that given vertices of A, B, C, D taken in order forms a parallelogram, it is sufficient to show that midpoint of AC is equals to midpoint of BD.

$\large\underline{\sf{Solution-}}$

Given coordinates are

• A (-2,-1)

• B (1,0)

• C (4,3)

and

• D (1,2)

We know,

Midpoint Formula :-

Let us consider a line segment joining the points A and B and let C (x, y) be the midpoint of AB, then coordinates of C is

$\boxed{ \quad\sf \:( x, y) = \bigg(\dfrac{x_1+x_2}{2} , \dfrac{y_1+y_2}{2} \bigg) \quad}$

$\sf \: where \: coordinates \: of \: A \: and \: B \: are \: (x_1,y_1) \: and \: B(x_2,y_2)$

Let us first find midpoint of AC.

• Coordinates of A = ( - 2, - 1)

• Coordinates of C = (4, 3)

Using midpoint Formula,

$\rm :\longmapsto\: \sf \: Midpoint \: of \: AC \: = \: \bigg(\dfrac{x_1+x_2}{2} , \dfrac{y_1+y_2}{2} \bigg)$

Here,

• • x₁ = - 2

• • x₂ = 4

• • y₁ = - 1

• • y₂ = 3

So,

$\rm :\longmapsto\:Midpoint \: of \: AC = \bigg(\dfrac{ - 2 + 4}{2} ,\dfrac{ - 1 + 3}{2} \bigg)$

$\rm :\longmapsto\:Midpoint \: of \: AC = \bigg(\dfrac{ 2}{2} ,\dfrac{2}{2} \bigg)$

$\rm :\longmapsto\:Midpoint \: of \: AC =(1 \: , \: 1)$

Now, To find Midpoint of BD,

• Coordinates of B = (1, 0)

• Coordinates of D = (1, 2)

Using Midpoint Formula,

$\rm :\longmapsto\:Midpoint \: of \: BD = \bigg(\dfrac{x_1+x_2}{2} , \dfrac{y_1+y_2}{2} \bigg)$

Here,

• • x₁ = 1

• • x₂ = 1

• • y₁ = 0

• • y₂ = 2

Thus,

$\rm :\longmapsto\:Midpoint \: of \: BD = \bigg(\dfrac{1 + 1}{2} ,\dfrac{2 + 0}{2} \bigg)$

$\rm :\longmapsto\:Midpoint \: of \: BD = \bigg(\dfrac{2}{2} ,\dfrac{2}{2} \bigg)$

$\rm :\longmapsto\:Midpoint \: of \: BD = (1, \: 1)$

$\bf\implies \:Midpoint \: of \: AC = Midpoint \: of \: BD$

$\bf\implies \:A,B,C,D \: are \: the \: vertices \: of \: a \: parallelogram.$

Additional Information :-

Distance Formula :-

Distance formula is used to find the distance between two given Points

${\underline{\boxed{\rm{\quad Distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \quad}}}}$

Section Formula :-

Section Formula is used to find the co ordinates of the point A(x, y) which divides the line segment joining the points (B) and (C) internally in the ratio m : n,

${\underline{\boxed{\rm{\quad \Big(x, y \Big) = \Bigg(\dfrac{mx_2 + nx_1}{m + n} ,\dfrac{my_2 + ny_1}{m + n}\Bigg) \quad}}}}$