## VERTICES OF RHOMBUS QUESTION1

Vertices of rhombus question1 :

Here we are going to see practice question on vertices of parallelogram.

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) 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.

## Vertices of rhombus question1 - 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 :

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

Since the lengths of all sides are equal, it may be a square or rhombus.

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

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

## Try other 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.

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