Question

Prove that the points (6, 8) (3, 7) (-2,-2), (1, -1) are the verticals of a parallelogram.​

Answer 1

Answer:

The given coordinates of the points are the vertices of a parallelogram.

Step-by-step-explanation:

NOTE: Refer to the attachment for the diagram.

Let the given points be A, B, C & D.

A ≡ ( 6, 8 ) ≡ ( x₁, y₁ )

B ≡ ( 3, 7 ) ≡ ( x₂, y₂ )

C ≡ ( - 2, - 2 ) ≡ ( x₃, y₃ )

D ≡ ( 1, - 1 ) ≡ ( x₄, y₄ )

Now, by Slope formula,

Slope of line AB = ( y₂ - y₁ ) / ( x₂ - x₁ )

⇒ Slope of line AB = ( 7 - 8 ) / ( 3 - 6 )

⇒ Slope of line AB = - 1 / - 3

⇒ Slope of line AB = 1 / 3 - - ( 1 )

Now,

Slope of line BC = ( y₃ - y₂ ) / ( x₃ - x₂ )

⇒ Slope of line BC = ( - 2 - 7 ) / ( - 2 - 3 )

⇒ Slope of line BC = - 9 / - 5

⇒ Slope of line BC = 9 / 5 - - ( 2 )

Now,

Slope of line CD = ( y₄ - y₃ ) / ( x₄ - x₃ )

⇒ Slope of line CD = [ - 1 - ( - 2 ) ] / [ 1 - ( - 2 ) ]

⇒ Slope of line CD = ( - 1 + 2 ) / ( 1 + 2 )

⇒ Slope of line CD = 1 / 3 - - ( 3 )

Now,

Slope of line AD = ( y₄ - y₁ ) / ( x₄ - x₁ )

⇒ Slope of line AD = ( - 1 - 8 ) / ( 1 - 6 )

⇒ Slope of line AD = - 9 / - 5

⇒ Slope of line AD = 9 / 5 - - ( 4 )

From ( 1 ) & ( 3 ),

Slope of line AB = Slope of line CD

∴ Line AB ∥ Line CD

From ( 2 ) & ( 4 ),

Slope of line BC = Slope of line AD

∴ Line BC ∥ Line AD

Now, in □ ABCD,

Opposite sides are parallel to each other.

∴ □ ABCD is a parallelogram. - - [ By definition ]

∴ The given coordinates of the points are the vertices of a parallelogram.

Hence proved!

Answer 2

Step-by-step explanation:

$\sf \bigg(AB = CD \& \bigg) \bigg(BC = AD \bigg) \\ \\ \\ \\ \\ \\ \sf \large \: AB = \sqrt{ \bigg(6 - 3 { \bigg)}^{2} + \bigg(8 - 7 { \bigg)}^{2} = }\sqrt{9 + 1} = \sqrt{10} \\ \\ \\ \\ \\ \\ \sf \large CD = \sqrt{ \bigg( - 2 - 1 { \bigg)}^{2} + \bigg( - 2 + 1 {\bigg)}^{2} } = \sqrt{9 + 1} = \sqrt{10} \\ \\ \\ \\ \\ \\ \sf \large BC = \sqrt{ \bigg( - 2 - 3 { \bigg)}^{2} + \bigg( - 2 - 7 { \bigg)}^{2} } = \sqrt{25 + 81} = \sqrt{106} \\ \\ \\ \\ \\ \\ \sf \large AD = \sqrt{ \bigg(6 - 1 { \bigg)}^{2} + \bigg(8 + 1 { \bigg)}^{2} } = \sqrt{81 + 25} = \sqrt{100} \\ \\ \\ \\ \\ \\ \bigg[ \sf \:AB = CD \& \: \: BC= AD \bigg] \\ \\ \\ \\ \\ \\ \sf \color{red} \large ABCD = parallelogram$