Question

In a classroom, 4 friends are seated at the points A,B,C,D as shown in figure. Champa and Chameli walk into class and observe 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?

Answer 1

Hint: Use distance formula. If all distances appear to be the same then it is definitely a square.
Distance between two points A(\[{x_{1,}}{y_1}\]) and B(\[{x_{2,}}{y_2}\]). In the bracket we have written coordinates.
\[d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \]

Complete step-by-step answer:
To know who is correct we have to find the distances.
Let’s first write coordinates of the points A,B,C and D.
A(3,4), B(6,7), C(9,4), D(6,1).
First is the x coordinate and second is the y coordinate.
We need to find distance AB,BC,CD,DA. These are four sides of that observed square.
So first find distance between AB.
\[d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \]
\[d(AB) = \sqrt {{{\left( {6 - 3} \right)}^2} + {{\left( {7 - 4} \right)}^2}} \] A(3,4) and B(6,7)
\[
   = \sqrt {{3^2} + {3^2}} \\
   = \sqrt {9 + 9} \\
   = \sqrt {18} \\
   = 3\sqrt 2 \\
\]
Similarly find distance between BC,CD and DA.
\[d(BC) = \sqrt {{{\left( {9 - 6} \right)}^2} + {{\left( {4 - 7} \right)}^2}} \] B(6,7) and C(9,4)
\[ = \sqrt {{3^2} + {{( - 3)}^2}} \\
   = \sqrt {9 + 9} \\
   = \sqrt {18} \\
   = 3\sqrt 2 \\
\]
\[d(CD) = \sqrt {{{\left( {6 - 9} \right)}^2} + {{\left( {1 - 4} \right)}^2}} \] C(9,4) and D(6,1)
\[
   = \sqrt {{{( - 3)}^2} + {{( - 3)}^2}} \\
   = \sqrt {9 + 9} \\
   = \sqrt {18} \\
   = 3\sqrt 2 \\
\]
\[d(DA) = \sqrt {{{\left( {3 - 6} \right)}^2} + {{\left( {4 - 1} \right)}^2}} \] D(6,1) and A(3,4)
\[
   = \sqrt {{{( - 3)}^2} + {3^2}} \\
   = \sqrt {9 + 9} \\
   = \sqrt {18} \\
   = 3\sqrt 2 \\
\]
Now we can see \[d(AB) = d(BC) = d(CD) = d(DA) = 3\sqrt 2 \]
This shows all four sides are equal . Thus concludes that Champa was right. ABCD is a square.

Note: First just locate the points and then write their coordinates.
Here instead of finding distance between four sides of a square ,we can also find the distance between opposite points forming diagonals. Because a rhombus is also having all sides the same but diagonals different.
Whereas a square has both diagonals of same length .
Diagonal of square=\[\sqrt 2 \times side\].
If distance between two opposite points=diagonal then it is also a proof for the figure to be a square.