If two opposite vertices of square are (1,2) and (5.8), find the coordinates of its other two vertices and the equations of its sides.

Explanation

Slope of `A C=frac{8-2}{6-1}=frac{3}{2}`

The sides AB and AD pass through the point `A(1,2)` and make an angle of `45^{circ}` with `A C` whose slope is `frac{3}{2}`.

Equations of AE and AD are given by

`y-2=frac{frac{2}{2}+tan 45^{y}}{1+frac{1}{2} tan 45^{2}}(x-1)`

`Rightarrow y-2=frac{3+2}{2+3}(x-1)`

`Rightarrow y-2=-5(x-1) text { and } y-2=frac{1}{5}(x-1)`

`Rightarrow 5 x+y-7=0 { and } x-5 y+9=0`

Thus, the equations of `A B` and `A D` are `5 x+y-7=0` and `x-5 y+9=0`. respectively.

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