# HOW TO PROVE THE GIVEN FOUR POINTS FORM A PARALLELOGRAM

## About "How to Prove the Given Four Points form a Parallelogram"

How to Prove the Given Four Points form a Parallelogram ?

Here we are going to see some example problems to show how to prove the given points form a parallelogram.

## How to Prove the Given Four Points form a Parallelogram - Practice questions

Question 1 :

Show that the following points taken in order form the vertices of a parallelogram.

(i) A(–3, 1), B(–6, –7), C (3, –9), D(6, –1)

Solution :

Distance between two points  =  √(x2 - x1)2 + (y2 - y1)2

A(–3, 1), B(–6, –7)

=  √(-6+3)2 + (-7-1)2

=  √(-3)2 + (-8)2

=  √(9 + 64)

AB   =  √73

B(–6, –7), C (3, –9)

=  √(3 + 6)2 + (-9 + 7)2

=  √92 + (-2)2

=  √(81 + 4)

BC  =  √85

C (3, –9), D(6, –1)

=  √(6 -3)2 + (-1 + 9)2

=  √32 + 82

=  √(9 + 64)

CD  =  √73

D(6, –1) A(–3, 1)

=  √(-3 - 6)2 + (1 + 1)2

=  √(-9)2 + 22

=  √(81 + 4)

DA  =  √85

Since opposite sides are having equal length it forms parallelogram.

(ii) A (–7, –3), B(5, 10), C(15, 8), D(3, –5)

Solution :

Distance between two points  =  √(x2 - x1)2 + (y2 - y1)2

A (–7, –3), B(5, 10)

=  √(5+7)2 + (10+3)2

=  √122 + 132

=  √(144 + 169)

AB   =  √313

B(5, 10), C(15, 8)

=  √(15 - 5)2 + (8 - 10)2

=  √102 + (-2)2

=  √(100 + 4)

BC  =  √104

C(15, 8), D(3, –5)

=  √(3 - 15)2 + (-5 - 8)2

=  √(-12)2 + (-13)2

=  √(144 + 169)

CD  =  √313

D(3, –5) A (–7, –3)

=  √(-7 - 3)2 + (-3 + 5)2

=  √(-10)2 + 22

=  √(100 + 4)

DA  =  √104

Since opposite sides are having equal length it forms parallelogram.

## How to Prove the Given Points for Rhombus - Questions

Question 1 :

Verify that the following points taken in order form the vertices of a rhombus.

(i) A(3, –2), B (7, 6),C (–1, 2), D (–5, –6)

Solution :

Distance between two points  =  √(x2 - x1)2 + (y2 - y1)2

A(3, –2), B (7, 6)

=  √(7-3)2 + (6+2)2

=  √42 + 82

=  √(16 + 64)

AB   =  √80

B (7, 6),C (–1, 2)

=  √(-1-7)2 + (2-6)2

=  √(-8)2 + (-4)2

=  √(64 + 16)

BC  =  √80

C (–1, 2), D (–5, –6)

=  √(-5+1)2 + (-6-2)2

=  √(-4)2 + (-8)2

=  √(16 + 64)

CD  =  √80

D(-5, -6) A(3, –2)

=  √(3 + 5)2 + (-2 + 6)2

=  √82 + 42

=  √(64 + 16)

DA  =  √80

Since all the sides are having same length, it forms a rhombus.

(ii) A (1, 1), B(2, 1),C (2, 2), D(1, 2)

Solution :

Distance between two points  =  √(x2 - x1)2 + (y2 - y1)2

A (1, 1), B(2, 1)

=  √(2-1)2 + (1-1)2

=  √12 + 02

AB   =  1

B(2, 1),C (2, 2)

=  √(2-2)2 + (2-1)2

=  √02 + 12

BC  =  1

C (2, 2), D(1, 2)

=  √(1-2)2 + (2-2)2

=  √(-1)2 + 02

CD  =  1

D(1, 2) A (1, 1)

=  √(1-1)2 + (1-2)2

=  √02 + 12

DA  =  1

Since all the sides are having same length, it forms a rhombus. After having gone through the stuff given above, we hope that the students would have understood, "How to Prove the Given Four Points form a Parallelogram"

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