# 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

**Solution:**

We use the distance formula and basic concepts of coordinate geometry to solve the problem given.

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: √{(x_{1} - x_{2})² + (y_{1} - y_{2})²}

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

- x
_{1}= 3 - y
_{1}= 4 - x
_{2}= 6 - y
_{1}= 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)

- x
_{1}= 6 - y
_{1}= 7 - x
_{2}= 9 - y
_{2}= 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)

- x
_{1}= 9 - y
_{1}= 4 - x
_{2}= 6 - y
_{2}= 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 B (3, 4) and D (6, 1)

- x
_{1}= 3 - y
_{1}= 4 - x
_{2}= 6 - y
_{1}= 1

AD = √{(3 - 6)² + (4 - 1)²}

= √(- 3)² + (3)²

= √9 + 9

= √18

= 3√2

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

- x
_{1}= 3 - y
_{1}= 4 - x
_{2}= 9 - y
_{2}= 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)

- x
_{1}= 6 - y
_{1}= 7 - x
_{2}= 6 - y
_{2}= 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

**Video Solution:**

## 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

### NCERT Class 10 Maths Solutions - Chapter 7 Exercise 7.1 Question 5:

If in a classroom, 4 friends are seated at points A, B, C, and D as shown in the above figure. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, hence 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.