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 told Chameli, “I think ABCD is a square”. State whether Champa's statement is correct or not.

Answer 1

Clearly from the figure, the coordinates of points A, B, C and D are (3, 4), (6, 7), (9, 4) and (6,1)
Distance between the points is given by
$\sqrt{{(x_1-x_2)}^2+{(y_1-y_2)}^2}$
So,
$AB=\sqrt{{(3-6)}^2+{(4-7)}^2}$
$=\sqrt{{(-3)}^2+{(-3)}^2}$
$=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}$
$BC=\sqrt{{(6-9)}^2+{(7-4)}^2}$
$=\sqrt{{(-3)}^2+3^2}$
$=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}$
$CD=\sqrt{{(9-6)}^2+{(4-1)}^2}$
$=\sqrt{{\left(3\right)}^2+{\left(3\right)}^2}$
$=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}$
$AD=\sqrt{{(3-6)}^2+{(4-1)}^2}$
$=\sqrt{{(-3)}^2+{\left(3\right)}^2}$
$=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}$
Diagonal $AC=\sqrt{{(3-9)}^2+{(4-4)}^2}$
$=\sqrt{{(-6)}^2+0^2}=6$
Diagonal $BD=\sqrt{{(6-6)}^2+{(7-1)}^2}$
$=\sqrt{0^2+{\left(6\right)}^2}=6$
It can be observed that all sides of this quadrilateral ABCD are of same length and also the diagonals are same length.
Therefore, ABCD is a square and hence, Champa is correct.