In a classroom, 4 friends are seated at the points A (3, 4), B (6, 7), C (9, 4) and D (6, 1). 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 distance formula, find which of them is correct.

We have A = (3, 4), B = (6, 7), C = (9, 4) and D = (6, 1)
Using Distance Formula to find distances AB, BC, CD and DA, we get

AB = \(\sqrt{(6-3)^2 + (7 - 4)^2} \)
\( = \sqrt{3^2 + 3^2} \)
\( = \sqrt{9 + 9} \)
\( = \sqrt{18} \)
\( = 3\sqrt{2}\)

BC = \(\sqrt{(9- 6)^2 + (4 - 7)^2} \)
\(= \sqrt{3^2 + 3^2} \)
\( = \sqrt{9 + 9} \)
\( = \sqrt{18} \)
\( = 3\sqrt{2}\)

CD = \(\sqrt{(6-9)^2 + (1 - 4)^2} \)
\( = \sqrt{3^2 + (-3)^2} = \sqrt{9 + 9} \)
\( = \sqrt{18} \)
\( = 3\sqrt{2}\)

DA = \(\sqrt{(6-3)^2 + (1 - 4)^2} \)
\( = \sqrt{3^2 + (-3)^2} \)
\( = \sqrt{9 + 9} \)
\( = \sqrt{18} \)
\( = 3\sqrt{2}\)

Therefore, All the sides of ABCD are equal here. … (1)

Now, we will check the length of its diagonals.

AC = \(\sqrt{(9-3)^2 + (4 - 4)^2} \)
\( = \sqrt{6^2 - 0^2} \)
\( = \sqrt{36} \)
\( = 6\)

BD = \(\sqrt{(6-6)^2 + (1 - 7)^2} \)
\( = \sqrt{0^2 - 6^2} \)
\( = \sqrt{36} \)
\( = 6\)

So, Diagonals of ABCD are also equal. … (2)

From (1) and (2), we can definitely say that ABCD is a square.

Therefore, Champa is correct.