#### Question

In a classroom, 4 friends are seated at the points A, B, C and D as shown in the following figure. 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

It can be observed that A (3, 4), B (6, 7), C (9, 4), and D (6, 1) are the positions of these 4 friends.

`AB = sqrt((3-6)^2+(4-7)^2) = sqrt((-3)^2+(-3)^2) = sqrt(9+9) = sqrt18 = 3sqrt2`

`BC = sqrt((6-9)^2+(7-4)^2) = sqrt((-3)^2+(3)^2) = sqrt(9+9) = sqrt18 = 3sqrt2`

`CD = sqrt((9-6)^2+(4-1)^2) = sqrt((3)^2+(3)^2) = sqrt(9+9) = sqrt18 =3sqrt2 `

`AD = sqrt((3-6)^2+(4-1)^2) = sqrt((-3)^2 + (3)^2) = sqrt(9+9) = sqrt18 = 3sqrt2`

Diagonal AC=`sqrt((3-9)^2+(4-4)^2)=sqrt((-6)^2)=6`

Diagonal BD=`sqrt((6-6)^2+(7-1)^2)=sqrt((6)^2)=6`

It can be observed that all sides of this quadrilateral ABCD are of the same length and also the diagonals are of the same length.

Therefore, ABCD is a square and hence, Champa was correct