Q.31. Children of a school planned to plant trees in their school. They planted tree in a pattern. Plot the following coordinates; A (- 2, 2), B (- 2, - 2), C (4, - 2) and D (4, 2). Join the points in order. What is the figure you get? Which values of the children depicted here?

Dear Student,

Please find below the solution to the asked query:

We plot given coordinates on a graph , As :

We know : Distance formula :  d = $\sqrt{{\left(x_2 - x_1\right)}^2 + {\left(y_2 - y_1\right)}^2 }$

Then, to find Length of AD , We get x₁ = - 2  , y₁ = 2  and  x₂ = 4  , y₂ = 2 

AD  = $\sqrt{{\left(4 - \left(- 2\right)\right)}^2 + {\left(2 - 2\right)}^2 } = \sqrt{{\left(4 +\hspace{0.17em}2\right)}^2 + 0^2 }= \sqrt{6^2}$ = 6 unit

And to find Length of BC , We get x₁ = - 2  , y₁ = - 2  and  x₂ = 4  , y₂ = - 2 

BC  = $\sqrt{{\left(4 - \left(- 2\right)\right)}^2 + {\left(\left(- 2\right) - \left(- 2\right)\right)}^2 } = \sqrt{{\left(4 +\hspace{0.17em}2\right)}^2 + {\left( - 2 +\hspace{0.17em}2\right)}^2 }= \sqrt{6^2}$ = 6 unit

So,

AD  =  BC                                                     --- ( 1 )

And we know equation of line when two coordinates on the line is given : ( y - y₁ ) = $\frac{y_{2 }- y_1}{x_2 - x_1}$ ( x - x₁ ) 
Then, to find equation of line AD , We get x₁ = - 2  , y₁ = 2  and  x₂ = 4  , y₂ = 2 

Equation of AD :

$$\begin{gathered} \left(y - 2\right) = \frac{2 - 2}{4 - \left( - 2\right)}\left(x - \left( - 2\right)\right) \\ \Rightarrow \left(y - 2\right) = \frac{0}{4 + 2}\left(x + 2\right) \\ \Rightarrow \left(y - 2\right) = 0 \\ \mathbf{\Rightarrow }\mathit{y}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{2} \end{gathered}$$
So, AD is parallel to ' x - axis '  .                                  --- ( 2 )


And to find equation of line BC , We get x₁ = - 2  , y₁ = - 2  and  x₂ = 4  , y₂ = - 2 

Equation of BC :

$$\begin{gathered} \left(y - \left( - 2\right)\right) = \frac{ \left( - 2\right) - \left( - 2\right)}{4 - \left( - 2\right)}\left(x - \left( - 2\right)\right) \\ \Rightarrow \left(y + 2\right) = \frac{- 2 + 2}{4 + 2}\left(x + 2\right) \\ \Rightarrow \left(y + 2\right) = \frac{0}{6}\left(x + 2\right) \\ \Rightarrow \left(y + 2\right) = 0 \\ \mathbf{\Rightarrow }\mathit{y}\mathbf{ }\mathbf{=}\mathbf{-}\mathbf{ }\mathbf{2} \end{gathered}$$
So, BC is parallel to ' x - axis '  .                                  --- ( 3 )

From statements ( 2 ) and ( 3 )  we get   AD | | BC  | | x - axis                           --- ( 4 )

Similarly we can show : AB | | CD  | | y - axis 

And we know x - axis and y axis are perpendiculars to each other , So

AD $\perp$ AB and AD $\perp$ CD  , BC $\perp$ AB and BC $\perp$ CD                                   --- ( 5 )

From equation , 1 4 and 5 we get  :


ABCD is a rectangle .

And by doing this work students depicted their caring nature towards environment .

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