## VERTICES OF SQUARE QUESTION5

Vertices of square question5 :

Here we are going to see how to check whether the given points form a square

Definition of square  :

Area enclosed by the 4 equal sides is called the area of the square.

## How to check if four points form a square ?

(i)  Plot the given points in the graph and draw the square.

(ii)  Now we need to find the length of all sides.

(iii)  In a square length of all sides are equal.

(iv)  Length of diagonals are also equal

If we prove the above two conditions(iii, iv) are true, we can decide that the given points form a square.

## Vertices of square question5 - Solution

Question 5 :

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

Solution :

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 (1,2) and B (2,2) and C (2,3) and D (1,3)

Distance between the points A and B

= √(x₂ - x₁)² + (y₂ - y₁)²

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

=    √(2-1)² + (2-2)²

=    √(1)² + (0)²

=    √1 + 0²

=    1 unit

Distance between the points B and C

= √(x₂ - x₁)² + (y₂ - y₁)²

Here x₁ = 2, y₁ = 2, x₂ = 2  and  y₂ = 3

=    √(2-2)² + (3-2)²

=    √(0)² + (1)²

=    √0² + 1

=   1 unit

Distance between the points C and D

= √(x₂ - x₁)² + (y₂ - y₁)²

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

=    √(1-2)² + (3-3)²

=    √(-1)² + (0)²

=    √1 + 0²

=    1 unit

Distance between the points D and A

= √(x₂ - x₁)² + (y₂ - y₁)²

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

=    √(1-1)² + (2-3)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.Now, 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 = 2, x₂ = 2  and  y₂ = 3

=    √(2-1)² + (3-2)²

=    √(1)² + 1²

=    √1 + 1

=    √2 units

Distance between the points B and D

= √(x₂ - x₁)² + (y₂ - y₁)²

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

=    √(1-2)² + (3-2)²

=    √(-1)² + (1)²

=    √1 + 1

=    √2 units

Length of all sides and diagonals are equal so the given vertices forms a square. Vertices of square question5

## Try other questions

(1)  Examine whether the given points  A (2,6) and B (5,1) and C (0,-2) and D (-3,3) forms a square.

(2)  Examine whether the given points  A (8,8) and B (13,-4) and C (1,-9) and D (-4,3) forms a square.

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

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

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

(6)  Examine whether the given points  A (-1,-8) and B (4,-6) and C (2,-1) and D (-3,-3) forms a square.

(7)  Examine whether the given points A (0,-1) and B (2,1) and C (0,3) and D (-2,1) forms a square.

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