HOW TO PROVE THE GIVEN FOUR POINTS FORM A PARALLELOGRAM

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.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.