Step 1: Understand the special distance. We do not want the usual perpendicular distance. We want the distance from $(1, 2)$ to the line, but measured along a particular direction parallel to $x - y = 0$. So we travel in that direction until we hit the line.
Step 2: Find the direction angle. The guide line $x - y = 0$ has slope $1$, which means it makes a $45^\circ$ angle. So $\cos 45^\circ = \frac{1}{\sqrt{2}}$ and $\sin 45^\circ = \frac{1}{\sqrt{2}}$.
Step 3: Write a moving point. Starting at $(1, 2)$ and moving a distance $r$ in that direction gives the point. \[ \left(1 + \frac{r}{\sqrt{2}},\ 2 + \frac{r}{\sqrt{2}}\right) \]
Step 4: Make it lie on the target line. This point must satisfy $3x + 4y - 32 = 0$. Substitute. \[ 3\left(1 + \frac{r}{\sqrt{2}}\right) + 4\left(2 + \frac{r}{\sqrt{2}}\right) - 32 = 0 \]
Step 5: Simplify the equation. Expand: $3 + \frac{3r}{\sqrt{2}} + 8 + \frac{4r}{\sqrt{2}} - 32 = 0$. Combine constants and the $r$ terms. \[ \frac{7r}{\sqrt{2}} - 21 = 0 \]
Step 6: Solve for $r$. Move and multiply. \[ \frac{7r}{\sqrt{2}} = 21 \implies r = \frac{21\sqrt{2}}{7} = 3\sqrt{2} \]
\[ \boxed{3\sqrt{2}} \]