G4.2 Given that $x$ is a real number and $y=\sqrt{x^{2}-2 x+2}+\sqrt{x^{2}-10 x+34}$. Find the minimum value of $y$. Reference: $\mathbf{2 0 1 5}$ HI9
Consider the following problem:
Let $P(1,1)$ and $Q(5,3)$ be two points. $R(x, 0)$ is a variable point on $x$-axis.
To find the minimum sum of distances $P R+R Q$.
Let $y=$ sum of distances $=\sqrt{(x-1)^{2}+1}+\sqrt{(x-5)^{2}+9}$
If we reflect $P(1,1)$ along $x$-axis to $P_{1}(1,-1), M(1,0)$ is the foot of perpendicular,
then $\triangle P M R \cong \triangle P_{1} M R$
$y=P R+R Q=P_{1} R+R Q \geq P_{1} Q$
$y \geq \sqrt{(5-1)^{2}+(3+1)^{2}}=4 \sqrt{2}$
(S.A.S.)
(triangle inequality)