Reflections, Midpoints and Distance

Reflecting a point is pure bookkeeping once you know the four rules: across the $x$-axis $(a,b) \to (a,-b)$; across the $y$-axis $(a,b) \to (-a,b)$; across $y = x$ the coordinates swap, $(a,b) \to (b,a)$; across $y = -x$ they swap and both negate, $(a,b) \to (-b,-a)$.

The midpoint of $(a,b)$ and $(c,d)$ is $\left(\frac{a+c}{2}, \frac{b+d}{2}\right)$ — average each coordinate separately.

For distance, work with the square first. $d^2 = (a-c)^2 + (b-d)^2$, and most contest questions ask for $d^2$ precisely so the radical never appears.