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 true or false.

• A. True
• B. False

Answer 1

The correct option is A True

Clearly from the figure, the coordinates of points A, B, C and D are (3, 4), (6, 7), (9, 4) and (6,1)
By using distance formula, we get:
$AB=\sqrt{{(3-6)}^2+{(4-7)}^2}=\sqrt{{(-3)}^2+{(-3)}^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}$
$BC=\sqrt{{(6-9)}^2+{(7-4)}^2}=\sqrt{{(-3)}^2+3^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}$
$CB=\sqrt{{(9-6)}^2+{(4-1)}^2}=\sqrt{{\left(3\right)}^2+{\left(3\right)}^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}$
$AD=\sqrt{{(3-6)}^2+{(4-1)}^2}=\sqrt{{(-3)}^2+{\left(3\right)}^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}$
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 the same length and also the diagonals are of the same length.
Therefore, ABCD is a square and hence, Champa is correct.