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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.


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Solution

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
(x1x2)2+(y1y2)2
So,
AB=(36)2+(47)2
=(3)2+(3)2
=9+9=18=32
BC=(69)2+(74)2
=(3)2+32
=9+9=18=32
CD=(96)2+(41)2
=(3)2+(3)2
=9+9=18=32
AD=(36)2+(41)2
=(3)2+(3)2
=9+9=18=32
Diagonal AC=(39)2+(44)2
=(6)2+02=6
Diagonal BD=(66)2+(71)2
=02+(6)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.

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