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Mathematics Question on Coordinate Geometry

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.
In a classroom, 4 friends are seated at points A, B, C and D.

Answer

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= (3−6)+2(4−7)2=(−3)2+(−3)2=9+9=18=32\sqrt{(3-6)+^2(4-7)^2}=\sqrt{(-3)^2+(-3)^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt2
BC= (6−9)+2(7−4)2=(−3)2+(3)2=9+9=18=32\sqrt{(6-9)+^2(7-4)^2}=\sqrt{(-3)^2+(3)^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt2
CB= (9−6)+2(4−1)2=(3)2+(3)2=9+9=18=32\sqrt{(9-6)+^2(4-1)^2}=\sqrt{(3)^2+(3)^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt2
AD= (3−6)+2(4−1)2=(−3)2+(3)2=9+9=18=32\sqrt{(3-6)+^2(4-1)^2}=\sqrt{(-3)^2+(3)^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt2
Diagonal AC=(3−9)2+(4−4)2=(−6)2+02=6\sqrt{(3-9)^2+(4-4)^2}=\sqrt{(-6)^2+0^2}=6
Diagonal BD=(6−6)2+(7−1)2=02+62=6\sqrt{(6-6)^2+(7-1)^2}=\sqrt{0^2+6^2}=6
4 friends are seated at points A, B, C and D
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.