The point (5, 2) is reflected in the straight line $y=x+2$. What are the coordinates of its image?

Reflecting in the line $y=x+c$ sends $(x,y)$ to $(y-c,x$ + c), so with $c=2$ the point (5, 2) becomes (0, 7). Both tests pass: the midpoint of the two points is (2.5, 4.5), which satisfies $y=x+2$, and the segment joining them has slope $(7-2)/(0-5)=-1$, at right angles to the mirror's slope of 1. The trap (2, 5) reflects in $y=x$ and forgets the shift altogether, (2, 7) reflects in $y=x$ and then slides the image up by 2 instead of applying the shift correctly, and (4, 3) reverses the two signs in the rule.