In a classroom, 4 friends are seated at the points A, B, C and D as shown in Fig. 7.8. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, “Don’t you think ABCD is a square?” Chameli disagrees. Using distance formula, find which of them is correct.

 

 

 

 

 

 

 

 

 

 

 

 

To prove that the points A, B, C, and D from a square, the length of the four sides should be equal and the length of the two diagonals should be the same.

Let A (3, 4), B (6, 7), C (9, 4), and D (6, 1) be the positions of 4 friends.

We know that the distance between the two points is given by the 

Distance Formula : √{(x₁ - x₂)² + (y₁ - y₂)²} 

To find AB, that is, the distance between points A (3, 4) and B (6, 7)

• x₁ = 3
• y₁ = 4
• x₂ = 6
• y₂ = 7

AB = √(3 - 6)² + (4 - 7)²

= √(- 3)² + (- 3)²

= √9 + 9

= √18

= 3√2

To find BC, that is, the distance between Points B (6, 7) and C (9, 4)

• x₁ = 6
• y₁ = 7
• x₂ = 9
• y₂ = 4

BC = √(6 - 9)² + (7 - 4)²

= √(- 3)² + (3)²

= √9 + 9

= √18

= 3√2

To find CD, that is, the distance between Points C (9, 4) and D (6, 1)

• x₁ = 9
• y₁ = 4
• x₂ = 6
• y₂ = 1

CB = √{(9 - 6)² + (4 - 1)²}

= √(3)² + (3)²

= √9 + 9

= √18

= 3√2

To find AD, that is, the distance between Points A (3, 4) and D (6, 1)

• x₁ = 3
• y₁ = 4
• x₂ = 6
• y₂ = 1

AD = √(3 - 6)² + (4 - 1)²

= √(- 3)² + (3)²

= √9 + 9

= √18

= 3√2

Now, let's find the length of the diagonals.

To find AC, that is, the distance between points A (3, 4) and C (9, 4)

• x₁ = 3
• y₁ = 4
• x₂ = 9
• y₂ = 4

Diagonal AC = √(3 - 9)² + (4 - 4)²

= √(- 6)² + 0²

= 6

To find BD, that is, the distance between Points B (6, 7) and D (6, 1)

• x₁ = 6
• y₁ = 7
• x₂ = 6
• y₂ = 1

Diagonal BD = √(6 - 6)² + (7 - 1)²

= √(0² + (6)²

= 6

The four sides AB, BC, CD, and AD are of the same length, and diagonals AC and BD are of equal length. Therefore, ABCD is a square and hence, Champa was correct.