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explain, why, on the coordinate plane, when symmetry about the bisector changes the coordinates of a point, (x,y) of the first point is (y,x) of the second point

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Let's look at how reflection axes work

The point you're reflecting, let's call it (X,Y)

Your reflection axis is a straight line, **0=ax+by+c**

In this specific case it's **0=1x-1y**

The direction **perpendicular** to this axis is **(1,-1)** which is 45° southeast

If (X,Y) moves into this direction a certain amount **K**, the point will land on the reflection axis itself

    (X,Y) + K·(1,-1)  =  (X+K, Y-K)
                         is on the axis

     0 = x - y
     0 = X+K - (Y-K)
     0 = X-Y + 2K
    2K = Y-X
     K = (Y-X)/2

But you want to move twice that amount so you get reflected to the other side

    (X,Y) + 2K·(1,-1)  =  (X+2K, Y-2K)
                          is on the other side

    Xnew = X + 2K           Ynew = Y - 2K
         = X + 2(Y-X)/2          = Y - 2(Y-X)/2
         = X + Y-X               = Y + X-Y
         = Y                     = X
by

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