To find the length of AO, we can use the Law of Cosines in triangle AOB. However, we first need to determine the side lengths and the angle between them.
Given:
- \( BO = 45 \) in (radius of the circle),
- \( BA = 108 \) in (distance from point A to B).
Let \( AO = x \). By the Law of Cosines,
\[ AB^2 = AO^2 + BO^2 - 2 \cdot AO \cdot BO \cdot \cos(\angle AOB) \]
Since no angle is provided, we might not need it if we observe the triangle could be positioned such that we can apply a more straightforward relation.
We also know that \( A, B, \) and \( O \) are points where \( A \) is outside the circle, \( B \) is on the circumference, and \( O \) is the center of the circle.
Using the triangle inequality,
\[ AB < AO + BO \] \[ 108 < x + 45 \] \[ x > 63 \]
And,
\[ AO < AB + BO \] \[ x < 108 + 45 \] \[ x < 153 \]
So \( x \) must be greater than 63 in and less than 153 in.
To find a numerical solution, we might also consider knowing that the triangle inequality forms boundaries and together with the lengths we derived above, we can determine \( AO \).
Using the approximate reference provided by the options, observe that:
If you visually test the responses:
- \( 117 \) in
- \( 67 \) in
- \( 98 \) in
- \( 25 \) in
Given the triangle inequality and distances from B to A, the most reasonable guess without explicit calculation can interpret as \( AO \) being closer to the maximum suggested response length.
Thus, choosing \( 117 \) in seems plausible considering the arrangement allows it to fulfill both calculated conditions of triangle inequalities.
The correct answer is:
117 in.