A ray leaves A(2, 2), strikes the mirror y = x + 3 at a point M, and then passes through B(5, 2)…

Step 1. Same side. $A$ gives $2 - 2 + 3 = 3$ and $B$ gives $5 - 2 + 3 = 6$. Both are positive, so $A$ and $B$ are on the same side of the mirror.

Step 2. The image of $B$. $t = \dfrac{5 - 2 + 3}{2} = 3$, so $B' = (5 - 2 \times 1 \times 3,\ 2 - 2 \times (-1) \times 3) = (-1, 8)$.

Step 3. The line $AB'$. Its slope is $\dfrac{8 - 2}{-1 - 2} = -2$, so $y = -2x + 6$.

Step 4. The point $M$. $x + 3 = -2x + 6$ gives $x = 1$ and $y = 4$.

Step 5. The length. $AB' = \sqrt{3^2 + 6^2} = \sqrt{45} \approx 6.708$.