Question

Reflect over the line y = x: (-2, -6)

Answer 1

The reflection in y=x interchanges the coordinates, so (-2,-6) becomes the reflection (-6,-2).

Here's one way to prove this. If P(a,b) is the point to be reflected as P'(a',b'), the line PP' is perpendicular to y=x so it has the equation y=-x+c, where c can be found by plugging in x=a, y=b, b=-a+c, c=a+b, and the equation of PP' is y=-x+a+b, which meets y=x when x=-x+a+b, so y=x=½(a+b). So the intersection point N has the coordinates (½(a+b),½(a+b)) and N is the midpoint of PP'. But the midpoint of PP' is N(½(a+a'),½(b+b')). Therefore ½(a+a')=½(a+b), a'=a+b-a=b; and ½(b+b')=½(a+b), b'=a+b-b=a. So P is (b,a).