**Vertices of parallelogram worksheet :**

Here we are going to see some
question to verify how to examine whether the given points are the
vertices of parallelogram.

(1) Examine whether the given points A (4,6) and B (7,7) and C (10,10) and D (7,9) forms a parallelogram.

(2) Examine whether the given points A (3,-5) and B (-5,-4) and C (7,10) and D (15,9) forms a parallelogram.

(3) Examine whether the given points A (-4,-3) and B (3,1) and C (3,6) and D (-4,2) forms a parallelogram.

(4) Examine whether the given points A (8,4) and B (1,3) and C (3,-1) and D (4,6) forms a parallelogram.

**Question 1 :**

Examine whether the given points A (4,6) and B (7,7) and C (10,10) and D (7,9) forms a parallelogram.

**Solution :**

To show that the given points are collinear we need to find the distance between three points.

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

**√(x₂ - x₁)² + (y₂ - y₁)²**

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

Distance between the points A and B

= **√(x₂ - x₁)² + (y₂ - y₁)²**

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

**= **√(7-4)² + (7-6)²

= √(3)² + (1)²

= √9 + 1 = √10

Distance between the points B and C

= **√(x₂ - x₁)² + (y₂ - y₁)²**

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

** = **√(10-7)² + (10-7)²

= √(3)² + (3)²

= √9 + 9

= √18 units

Distance between the points C and D

= **√(x₂ - x₁)² + (y₂ - y₁)²**

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

** = **√(7-10)² + (9-10)²

= √(-3)² + (-1)²

= √9 + 1

= √10 units

Distance between the points D and A

= **√(x₂ - x₁)² + (y₂ - y₁)²**

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

** = **√(4-7)² + (6-9)²

= √(-3)² + (-3)²

= √9 + 9

= √18 units

AB = √10 units

BC = √18 units

CD = √10 units

DA = √18 units

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

Let us see the next problems on "Vertices of parallelogram worksheet" "Vertices of parallelogram worksheet"

**Question 2 :**

Examine whether the given points A (3,-5) and B (-5,-4) and C (7,10) and D (15,9) forms a parallelogram.

**Solution :**

To show that the given points forms a rectangle we need to find the distance between three points.

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

**√(x₂ - x₁)² + (y₂ - y₁)²**

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

Distance between the points A and B

= **√(x₂ - x₁)² + (y₂ - y₁)²**

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

**= ** √(-5-3)² + (-4-(-5))²

= √(-8)² + (-4+5)²

=** **√64 + 1²

= √(64 + 1)

= √65 units

Distance between the points B and C

= **√(x₂ - x₁)² + (y₂ - y₁)²**

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

Distance between the points C and D

= **√(x₂ - x₁)² + (y₂ - y₁)²**

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

**= **√(15-7)² + (9-10)²

= √(8)² + (-1)²

= ** **√64 + 1

= √65 units

Distance between the points D and A

= **√(x₂ - x₁)² + (y₂ - y₁)²**

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

**= **√(3-15)² + (-5-9)²

= √(-12)² + (-14)²

= √144 + 196

= √340 units

AB = √65 units

BC = √340 units

CD = √65 units

DA = √340 units

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

Let us see the next problems on "Vertices of parallelogram worksheet"

**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 :**

To show that the given points forms a rectangle we need to find the distance between three points.

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

**√(x₂ - x₁)² + (y₂ - y₁)²**

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

Distance between the points A and B

= **√(x₂ - x₁)² + (y₂ - y₁)²**

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

Distance between the points B and C

= **√(x₂ - x₁)² + (y₂ - y₁)²**

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

**= **√(3-3)² + (6-1)²

= √(0)² + (5)²

= ** **√0 + 25

= √25 units

Distance between the points C and D

= **√(x₂ - x₁)² + (y₂ - y₁)²**

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

**= ** √(-4-3)² + (2-6)²

= √(-7)² + (-4)²

= √49 + 16

= √65 units

Distance between the points D and A

= **√(x₂ - x₁)² + (y₂ - y₁)²**

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

**= **√(-4-(-4))² + (-3-2)²

= √(-4+4)² + (-5)²

= √0 + 25

= √25 units

AB = √65 units

BC = √25 units

CD = √65 units

DA = √25 units

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

Let us see the next problems on "Vertices of parallelogram worksheet"

**Question 4 :**

Examine whether the given points A (8,4) and B (1,3) and C (3,-1) and D (4,6) forms a parallelogram.

**Solution :**

To show that the given points forms a rectangle we need to find the distance between three points.

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

**√(x₂ - x₁)² + (y₂ - y₁)²**

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

Distance between the points A and B

= **√(x₂ - x₁)² + (y₂ - y₁)²**

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

**= **√(1-8)² + (3-4)²

= √(-7)² + (-1)²

= √49 + 1

= √50 units

Distance between the points B and C

= **√(x₂ - x₁)² + (y₂ - y₁)²**

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

**= **√(3-1)² + (-1-3)²

= √(2)² + (-4)²

= √4 + 16

= √20 units

Distance between the points C and D

= **√(x₂ - x₁)² + (y₂ - y₁)²**

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

**= **√(4-3)² + (6-(-1))²

= √(1)² + (6+1)²

= √1 + 7²

= √1 + 49

= √50 units

Distance between the points D and A

= **√(x₂ - x₁)² + (y₂ - y₁)²**

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

**= **√(8-4)² + (4-6)²

= √(4)² + (-2)²

= ** **√16 + 4

= √20 units

AB = √50 units

BC = √20 units

CD = √50 units

DA = √20 units

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

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