HOW TO CHECK IF THE GIVEN FOUR POINTS FORM A RHOMBUS

How to Check if the Given Four Points Form a Rhombus ?

Here we are going to see some example problems to know how to examine if the given points form a rhombus.

To verify if the given four points form a rhombus, we need to follow the steps given below.

(i) Find the length of all sides using the formula distance between two points.

(ii) In any square the length of diagonal will be equal, to prove the given shape is not square but a rhombus, we need to prove that length of diagonal are not equal.

Verifying If the Given Four Points Form a Rhombus - Examples

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 :

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 :

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 :

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 :

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 :

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 diagonal AC :

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

Length of diagonal BD :

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

=    √(-6-6)² + (-7-5)²

=    √(-12)² + (-12)²

=    √144 + 144

=    √288 units

Since the lengths of diagonals are not equal, the given vertices will form a rhombus.

Try other questions

(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

(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

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