VERTICES OF RHOMBUS WORKSHEET
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.
How to test whether the given points are vertices of rhombus ?
(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.
Vertices of rhombus worksheet - Practice questions
(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.
Vertices of rhombus worksheet - Solution
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.
Related topics
- 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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