**Vertices of rhombus worksheet :**

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

**Definition of Rhombus :**

A quadrilateral which is having four equal sides is called rhombus. In other words a parallelogram will become rhombus if the diagonals are perpendicular.

(i) First we have to find the length of all sides using distance between two points formula.

(ii) Length of diagonal must be equal in any square.So we need to prove this the length of diagonals are not same.If the given vertices are satisfying these conditions then we can say the given vertices forms a rhombus.

(1) Examine whether the given points A (2,-3) and B (6,5) and C (-2,1) and D (-6,-7) forms a rhombus.

(2) Examine whether the given points A (1,4) and B (5,1) and C (1,-2) and D (-3,1) forms a rhombus.

(3) Examine whether the given points A (1,1) and B (2,1) and C (2,2) and D (1,2) forms a rhombus.

**Question 1 :**

Examine whether the given points A (2,-3) and B (6,5) and C (-2,1) and D (-6,-7) forms a rhombus.

**Solution : **Vertices of rhombus worksheet

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

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

**√(x₂ - x₁)² + (y₂ - y₁)²**

Four points are A (2,-3) and B (6,5) and C (-2,1) and D (-6,-7)

Distance between the points A and B

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = 2, y₁ = -3, x₂ = 6 and y₂ = 5**

**= ** √(6-2)² + (5-(-3))²

= √(4)² + (5+3)²

= ** **√16 + 8²

= ** **√16 + 64

= √80 units

Distance between the points B and C

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = 6, y₁ = 5, x₂ = -2 and y₂ = 1**

**= ** √(-2-6)² + (1-5)²

= √(-8)² + (-4)²

= ** **√64 + 16

= √80 units

Distance between the points C and D

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = -2, y₁ = 1, x₂ = -6 and y₂ = -7**

**= ** √(-6-(-2))² + (-7-1)²

= √(-6+2)² + (-8)²

= ** **√(-4)² + 64

= √16 + 64

= √80 units

Distance between the points D and A

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = -6, y₁ = -7, x₂ = 2 and y₂ = -3**

**= ** √(2-(-6))² + (-3-(-7))²

= √(2+6)² + (-3+7)²

= ** **√8² + 4²

= √64 + 16

= √80 units

AB = √80 units

BC = √80 units

CD = √80 units

DA = √80 units

Length of opposite sides are equal.To test whether it forms right triangle we need to find the length of diagonal AC and BD.

Distance between the points A and C

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = 2, y₁ = -3, x₂ = -2 and y₂ = 1**

**= ** √(-2-2)² + (1-(-3))²

= √(-4)² + (1+3)²

= ** **√16 + 4²

= ** **√16 + 16

= √32 units

Distance between the points B and D

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = 6, y₁ = 5, x₂ = -6 and y₂ = -7**

**= ** √(-6-6)² + (-7-5)²

= √(-12)² + (-12)²

= ** **√144 + 144

= √288 units

Length of all sides and diagonals are not equal so the given vertices will not form a rhombus.

Let us see solution of next problem on "Vertices of rhombus worksheet".

**Question 2 :**

Examine whether the given points A (1,4) and B (5,1) and C (1,-2) and D (-3,1) forms a rhombus.

**Solution : **Vertices of rhombus worksheet

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

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

**√(x₂ - x₁)² + (y₂ - y₁)²**

Four points are A (1,4) and B (5,1) and C (1,-2) and D (-3,1)

Distance between the points A and B

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = 1, y₁ = 4, x₂ = 5 and y₂ = 1**

**= ** √(5-1)² + (1-4)²

= √(4)² + (-3)²

= ** **√16 + 9

= √25

= 5 units

Distance between the points B and C

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = 5, y₁ = 1, x₂ = 1 and y₂ = -2**

**= ** √(1-5)² + (-2-1)²

= √(-4)² + (-3)²

= ** **√16 + 9

= ** **√25

= 5 units

Distance between the points C and D

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = 1, y₁ = -2, x₂ = -3 and y₂ = 1**

**= ** √(-3-1)² + (1-(-2))²

= √(-4)² + (1+2)²

= ** **√16 + 3²

= √25

= 5 units

Distance between the points D and A

= **√(x₂ - x₁)² + (y₂ - y₁)²**

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

**= ** √(1-(-3))² + (4-1)²

= √(1+3)² + (3)²

= ** **√4² + 9

= √16 + 9

= √25

= 5 units

AB = 5 units

BC = 5 units

CD = 5 units

DA = 5 units

Length of opposite sides are equal.To test whether it forms right triangle we need to find the length of diagonal AC and BD.

Distance between the points A and C

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = 1, y₁ = 4, x₂ = 1 and y₂ = -2**

**= ** √(1-1)² + (-2-4)²

= √(0)² + (-6)²

= ** **√0 + 36

= ** **√36

= 6 units

Distance between the points B and D

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = 5, y₁ = 1, x₂ = -3 and y₂ = 1**

**= ** √(-3-5)² + (1-1)²

= √(-8)² + (0)²

= ** **√64

= 8 units

Length of all sides and diagonals are not equal so the given vertices will not form a rhombus.

Let us see solution of next problem on "Vertices of rhombus worksheet".

**Question 3 :**

Examine whether the given points A (1,1) and B (2,1) and C (2,2) and D (1,2) forms a rhombus.

**Solution : **Vertices of rhombus worksheet

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

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

**√(x₂ - x₁)² + (y₂ - y₁)²**

Four points are A (1,1) and B (2,1) and C (2,2) and D (1,2)

Distance between the points A and B

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = 1, y₁ = 1, x₂ = 2 and y₂ = 1**

**= ** √(2-1)² + (1-1)²

= √(1)² + (0)²

= ** **√1

= 1 unit

Distance between the points B and C

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = 2, y₁ = 1, x₂ = 2 and y₂ = 2**

**= ** √(2-2)² + (2-1)²

= √(0)² + (1)²

= ** **√0 + 1

= 1 unit

Distance between the points C and D

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = 2, y₁ = 2, x₂ = 1 and y₂ = 2**

**= ** √(1-2)² + (2-2)²

= √(-1)² + (0)²

= ** **√(1) + 0

= 1 unit

Distance between the points D and A

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = 1, y₁ = 2, x₂ = 1 and y₂ = 1**

**= ** √(1-1)² + (1-2)²

= √(0)² + (-1)²

= ** **√0 + 1

= 1 unit

AB = 1 unit

BC = 1 unit

CD = 1 unit

DA = 1 unit

Length of opposite sides are equal.To test whether it forms right triangle we need to find the length of diagonal AC and BD.

Distance between the points A and C

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = 1, y₁ = 1, x₂ = 2 and y₂ = 2**

**= ** √(2-1)² + (2-1)²

= √(1)² + (1)²

= ** **√1 + 1

= √2 units

Distance between the points B and D

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = 2, y₁ = 1, x₂ = 1 and y₂ = 2**

**= ** √(1-2)² + (2-1)²

= √(-1)² + (1)²

= ** **√1 + 1

= √2 units

Length of all sides and diagonals are not equal so the given vertices will not form a rhombus.

- How to check if the given points are vertices of a right triangle ?
- How to know if the points are vertices of a parallelogram ?
- How to check if the given points are vertices of square ?

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