Question

In a classroom, 4 friends are seated at points A, B, C and D. 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 the distance formula, find which of them is correct.

Answer 1

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= \(\sqrt{(3-6)+^2(4-7)^2}=\sqrt{(-3)^2+(-3)^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt2\)
BC= \(\sqrt{(6-9)+^2(7-4)^2}=\sqrt{(-3)^2+(3)^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt2\)
CB= \(\sqrt{(9-6)+^2(4-1)^2}=\sqrt{(3)^2+(3)^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt2\)
AD= \(\sqrt{(3-6)+^2(4-1)^2}=\sqrt{(-3)^2+(3)^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt2\)
Diagonal AC=\(\sqrt{(3-9)^2+(4-4)^2}=\sqrt{(-6)^2+0^2}=6\)
Diagonal BD=\(\sqrt{(6-6)^2+(7-1)^2}=\sqrt{0^2+6^2}=6\)

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.