10. Design a right-angle ruler $ABC, BC$ on the ground, $AB$ perpendicular to the ground, and $AB=10 \mathrm{~cm}$. Move a wheel with a radius of no less than $10 \mathrm{~cm}$, so that the wheel is tightly against point $A$, and is tangent to $BC$ at point $D$ (as shown in Figure 2). The design requirement is that the scale at point $D$ should exactly show the radius of this wheel (in $\mathrm{cm}$ units). Therefore, when the length of $BC$ is $1 \mathrm{~m}$, the maximum scale that can be marked on $BC$ is $\qquad$
10.505.
$$
A B=10, B C=100 \text {. }
$$

Let $O A=O D=r$.
When $D$ coincides with $C$, as shown in Figure 6, the radius $r$ is at its maximum, which is the maximum scale marked, at this time,
$$
O H=r-10, A H=B C=100,
$$

Therefore, $100^{2}+(r-10)^{2}=r^{2}$.
Solving for $r$ gives $r=505$.