VERIFY THE GIVEN POINTS ARE VERTICES OF PARALLELOGRAM WORKSHEET

Examine whether the given points forms a parallelogram.

(i)  A(4, 6), B(7, 7) C(10, 10) and D (7, 9)

(ii)  A(3, -5), B(-5, -4), C (7, 10) and D (15, 9)

(iii)  A (-4, -3) and B (3, 1) and C (3, 6) and D (-4, 2)

(iv)  A(8, 4), B(1, 3), C(3, -1) and D (4, 6)

Detailed Solution

Question 1 :

A(4, 6), B(7, 7) C(10, 10) and D (7, 9)

Solution :

Distance Between Two Points (x ₁, y₁) and (x₂ , y₂)

√(x₂ - x₁)² + (y₂ - y₁)²

Length of AB :

Here x₁ = 4, y₁ = 6, x₂ = 7  and  y₂ = 7

=  √(7-4)² + (7-6)²

=  √3²+1²

=  √10

Length of BC :

Here x₁ = 7, y₁ = 7, x₂ = 10  and  y₂ = 10

 =  √(10-7)² + (10-7)²

  =  √3²+3²

  =  √(9+9)

  =  √18 units

Length of CD :

Here x₁ = 10, y₁ = 10, x₂ = 7  and  y₂ = 9

 =  √(7-10)²+(9-10)²

  =  √(-3)²+(-1)²

  =  √(9+1)

  =  √10 units

Length of DA :

Here x₁ = 7, y₁ = 9, x₂ = 4 and  y₂ = 6

 = √(4-7)²+(6-9)²

  =  √(-3)²+(-3)²

  =  √(9+9)

  =  √18 units

Length of opposite sides are equal. So, the given vertices will form a parallelogram.

Question 2 :

Examine whether the given points 

A(3, -5), B(-5, -4), C (7, 10) and D (15, 9)

forms a parallelogram.

Solution :

Length of AB :

Here x₁ = 3, y₁ = -5, x₂ = -5 and  y₂ = -4

=  √(-5-3)² + (-4-(-5))²

=  √(-8)²+(-4+5)²

=  √(64+1)

=  √65 units

Length of BC :

Here x₁ = -5, y₁ = -4, x₂ = 7 and  y₂ = 10

=  √(7-(-5))²+(10-(-4))²

=   √(7+5)²+(10+4)²

=   √12²+14²

=   √(144+196)

=  √340 units

Length of CD :

Here x₁ = 7, y₁ = 10, x₂ = 15 and  y₂ = 9

=  √(15-7)²+(9-10)²

=  √8²+(-1)²

=  √(64+1)

=  √65 units

Length of DA :

Here x₁ = 15, y₁ = 9, x₂ = 3 and  y₂ = -5

=  √(3-15)²+(-5-9)²

=  √(-12)²+(-14)²

=  √(144+196)

=  √340 units

Length of opposite sides are equal. So, the given vertices will form a parallelogram.

Question 3 :

Examine whether the given points

A (-4, -3) and B (3, 1) and C (3, 6) and D (-4, 2)

forms a parallelogram.

Solution :

Length of AB :

Here x₁ = -4, y₁ = -3, x₂ = 3 and  y₂ = 1

=  √(3-(-4))²+(1-(-3))²

=   √(3+4)²+(1+3)²

=  √7²+4²

=  √(49+16)

=  √65 units

Length of BC :

Here x₁ = 3, y₁ = 1, x₂ = 3 and  y₂ = 6

=  √(3-3)² + (6-1)²

=   √5²

=  5 units

Length of CD :

Here x₁ = 3, y₁ = 6, x₂ = -4 and  y₂ = 2

=  √(-4-3)² + (2-6)²

=   √(-7)² + (-4)²

=  √(49+16)

=  √65 units

Length of DA :

Here x₁ = -4, y₁ = 2, x₂ = -4 and  y₂ = -3

=  √(-4-(-4))² + (-3-2)²

=   √(-4+4)² + (-5)²

=  √25

=  5 units

Length of opposite sides are equal. So, the given vertices will form a parallelogram.

Question 4 :

Examine whether the given points 

A(8, 4), B(1, 3), C(3, -1) and D (4, 6)

forms a parallelogram.

Solution :

Length of AB :

Here x₁ = 8, y₁ = 4, x₂ = 1 and  y₂ = 3

=  √(1-8)² + (3-4)²

=   √(-7)² + (-1)²

=  √(49+1)

=  √50 units

Length of BC :

Here x₁ = 1, y₁ = 3, x₂ = 3 and  y₂ = -1

=  √(3-1)² + (-1-3)²

=   √(2)²+(-4)²

=  √(4+16)

=  √20 units

Length of CD :

Here x₁ = 3, y₁ = -1, x₂ = 4 and  y₂ = 6

=  √(4-3)² + (6-(-1))²

=  √1²+(6+1)²

=  √1+7²

=  √(1+49)

=  √50 units

Length of DA :

Here x₁ = 4, y₁ = 6, x₂ = 8 and  y₂ = 4

=  √(8-4)²+(4-6)²

=  √4²+(-2)²

=  √(16+4)

=  √20 units

Length of opposite sides are equal. So the given vertices forms a parallelogram

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