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Question

4 friends are seated at points A, B, C, and D in a classroom, as shown in the following figure. Amy walked into the class, and after observing for a few minutes, she thought, “ABCD must be a square”. State whether Amy’s statement is correct or not.


A
No
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B
None of the above
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C
Yes
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D
Can't be determined
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Solution

The correct option is C Yes

Clearly from the figure, the coordinates of points A, B, C and D are (3, 4), (6, 7), (9, 4) and (6,1).

We know that, the distance between the points (x1,y1) and (x2,y2) is represented as:

Distance, d=(x2x1)2+(y2y1)2

So, the distance of all the points given on the graph, are:

AB=(63)2+(74)2

=(3)2+(3)2

=9+9=18=32


BC=(96)2+(47)2

=(3)2+(3)2

=9+9=18=32


CD=(69)2+(14)2

=(3)2+(3)2

=9+9=18=32



DA=(63)2+(14)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 the quadrilateral ABCD are of same length and also the diagonals have same length.


So, ABCD is a square and hence, Amy is correct.

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