For every point in the plane (x, y), a 90° rotation can be described by the transformation P(x, y) → P'(-y, x). We can achieve this same transformation by performing two reflections.
A reflection across the line y = x "swaps" the coordinates of every point so that every point P(x, y) transforms into a new point P'(y, x). If we follow this with a reflection across the y-axis, we can flip the sign of our x-coordinate, resulting in a new point P''(-y, x). To review:
[tex] P(x,y)\xrightarrow[y=x]{reflect}P'(y,x)\xrightarrow[y-axis]{reflect}P''(-y,x) [/tex]
comparing this to the effect of a 90° rotation:
[tex] P(x,y)\xrightarrow[90^{\circ}]{rotate}P'(-y,x) [/tex]
We can see that the results are identical, so reflecting a figure across the line y = x and then across the y-axis is equivalent to rotating it 90° counterclockwise.
