Answer: \(P=(-4,-9)\) and \(R=(16,-1)\).

Use gradients for straight-line work: parallel lines have equal gradients and perpendicular lines have gradients whose product is \(-1\).

The midpoint of \(AB\) is

\(M=\left(\frac{2+10}{2},\frac{5+(-15)}{2}\right)=(6,-5).\)

The gradient of \(AB\) is

\(\frac{-15-5}{10-2}=-\frac{20}{8}=-\frac52.\)

Therefore the gradient of the perpendicular bisector is

\(\frac25.\)

The perpendicular bisector passes through \(M=(6,-5)\), so its equation is

\(y+5=\frac25(x-6).\)

Since \(P\) has \(y=-9\), substitute \(y=-9\):

\(-9+5=\frac25(x-6).\)

So

\(-4=\frac25(x-6).\)

Hence

\(x-6=-10,\)

and

\(x=-4.\)

Thus

\(P=(-4,-9).\)

If \(R\) is the reflection of \(P\) in \(AB\), then the midpoint of \(PR\) lies on \(AB\) at \(M\). Therefore

\(R=2M-P.\)

So

\(R=(12,-10)-(-4,-9)=(16,-1).\)