If P (-4, k), Q (k, 5), R (2, k) and S (k, 5) are the coordinates of a rectangle, find the value of k. Also find the coordinates

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Please find below the solution to the asked query:

Given : If P (-4, k), Q (k, 5), R (2, k) and S (k, 5) are the coordinates of a rectangle,


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

Now we find length of PQ , So x₁ =  - 4 , x₂ =  k  and  y₁ = k , y₂ = 5 , Then ,

PQ  = $\sqrt{{\left( \mathrm{k} - \left( - 4\right)\right)}^2 + {\left( 5 - \mathrm{k}\right)}^2} =\sqrt{{\left( \mathrm{k} + 4 \right)}^2 + {\left( 5 - \mathrm{k}\right)}^2}= \sqrt{{\mathrm{k}}^2 +\hspace{0.17em}16 + 8 \mathrm{k} + 25 + {\mathrm{k}}^2 - 10 \mathrm{k}} =\sqrt{\mathbf{2}{\mathbf{k}}^{\mathbf{2}}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{2}\mathbf{ }\mathbf{k}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{41}} \mathbf{unit}$

Now we find length of  RS , So x₁ = 2 , x₂ =  k  and  y₁ = k , y₂ = 5 , Then ,

RS  = $\sqrt{{\left( \mathrm{k} - 2\right)}^2 + {\left( 5 - \mathrm{k}\right)}^2} = \sqrt{{\mathrm{k}}^2 +\hspace{0.17em}4 - 4 \mathrm{k} + 25 + {\mathrm{k}}^2 - 10 \mathrm{k}} =\sqrt{\mathbf{2}{\mathbf{k}}^{\mathbf{2}}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{14}\mathbf{ }\mathbf{k}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{29}} \mathbf{unit}$

We know opposite sides are equal in length in rectangle , So 

PQ  =  RS , Substitute values we get :

$$\begin{gathered} \sqrt{2{\mathrm{k}}^2 - 2 \mathrm{k} + 41} = \sqrt{2{\mathrm{k}}^2 - 14 \mathrm{k} + 29} \\ \Rightarrow 2{\mathrm{k}}^2 - 2 \mathrm{k} + 41 = 2{\mathrm{k}}^2 - 14 \mathrm{k} + 29 \\ \Rightarrow 12 \mathrm{k} = - 12 \\ \mathbf{\Rightarrow }\mathbf{ }\mathbf{k}\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{1}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{Ans}\mathbf{ }\mathbf{)} \end{gathered}$$
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