To find the length of AO using the Pythagorean theorem, we can consider triangle AOB, where:
- BO is one leg of the triangle = 45 inches
- BA is the other leg of the triangle = 108 inches
- AO is the hypotenuse of the triangle
According to the Pythagorean theorem:
\[ AO^2 = AB^2 + BO^2 \]
Substituting the values we have:
\[ AO^2 = 108^2 + 45^2 \]
Calculating each square:
\[ 108^2 = 11664 \] \[ 45^2 = 2025 \]
Now, sum the squares:
\[ AO^2 = 11664 + 2025 = 13689 \]
Taking the square root to find AO:
\[ AO = \sqrt{13689} \approx 117 \]
Thus, the length of AO to the nearest whole inch is 117 inches.