To solve for the length of segment \( AO \), we can use the given lengths \( BO \) and \( BA \) to determine \( AO \).
We know that:
- \( BO = 45 \) inches
- \( BA = 108 \) inches
We can find the length of \( AO \) using the relationship:
\[ BA = BO + AO \]
Substituting the known values:
\[ 108 = 45 + AO \]
To find \( AO \), subtract \( 45 \) from both sides:
\[ AO = 108 - 45 \] \[ AO = 63 \text{ inches} \]
Therefore, the length of \( AO \) to the nearest whole inch is:
\[ \boxed{63} \]