Question

In a classroom, $ 4$ friends are seated at the points $ A,B,C$ and $ D$ as shown in Fig. 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.

Columns

Answer 1

Step $1:$ Describe the figure.

From the given figure the coordinates of $A,B,C$ and $D$ are $(3,4),(6,7),(9,4)$ and $(6,1)$ respectively.

Step $2:$ Find the distance $AB$ and $BC$.

As we know, according to the distance formula, the distance between two points $(x_1,y_1),(x_2,y_2)$ is given by $\sqrt{{(x_2-x_1)}^2+{\left(y_2-y_1\right)}^2}$.

Find the distance between point $A$ and point $B$.

$$\begin{gathered} AB=\sqrt{{(6-3)}^2+{\left(7-4\right)}^2} \\ \Rightarrow AB=\sqrt{{\left(3\right)}^2+{\left(3\right)}^2} \\ \Rightarrow AB=\sqrt{18} \\ \Rightarrow AB=3\sqrt{2} \end{gathered}$$

Therefore, the distance between point $A$ and point $B$ is $3\sqrt{2}$.

Find the distance between point $B$ and point $C$.

$$\begin{gathered} BC=\sqrt{{(9-6)}^2+{\left(4-7\right)}^2} \\ \Rightarrow BC=\sqrt{{\left(3\right)}^2+{\left(-3\right)}^2} \\ \Rightarrow BC=\sqrt{18} \\ \Rightarrow BC=3\sqrt{2} \end{gathered}$$

Therefore, the distance between point $B$ and point $C$ is $3\sqrt{2}$.

Step $3:$ Find the distance $CD$ and $AD$.

Find the distance between point $C$ and point $D$.

$$\begin{gathered} CD=\sqrt{{(6-9)}^2+{\left(1-4\right)}^2} \\ \Rightarrow CD=\sqrt{{(-3)}^2+{\left(-3\right)}^2} \\ \Rightarrow CD=\sqrt{18} \\ \Rightarrow CD=3\sqrt{2} \end{gathered}$$

Therefore, the distance between point $C$ and point $D$ is $3\sqrt{2}$.

Find the distance between point $A$ and point $D$.

$$\begin{gathered} AD=\sqrt{{(6-3)}^2+{\left(1-4\right)}^2} \\ \Rightarrow AD=\sqrt{{\left(3\right)}^2+{\left(-3\right)}^2} \\ \Rightarrow AD=\sqrt{18} \\ \Rightarrow AD=3\sqrt{2} \end{gathered}$$

Therefore, the distance between point $A$ and point $D$ is $3\sqrt{2}$.

Since, $AB,BC,CD$ and $AD$ all are of equal length.

Therefore, $ABCD$ is a square.

Hence, Champa is correct.