Past exam of the options I for mathematics and philosophy course of the University of Oxford / 2013 / B / 1 / c

First use the preceding isometry $t$ to align the first two points, and put $R_3=t(P_3)$. It remains to align $R_3$ with $Q_3$ while fixing $Q_1,Q_2$. Use orthonormal coordinates in which $Q_1=(0,0)$ and $Q_2=(d,0)$, $d>0$, and write $R_3=(r,s)$, $Q_3=(u,v)$. The prescribed distances giveSubtracting yields $r=u$, and then $s^2=v^2$. Therefore $Q_3$ is either $R_3$ or its reflection across the line through $Q_1,Q_2$. In the first case take $s=t$; in the second compose $t$ with that reflection. Both constructions fix the first two images and send the third point to its required image. The argument also covers three collinear points, when the two possible third images coincide. Thus all three points can be matched by one plane isometry.