To solve this problem, we start with the initial segment \( AB \), which is 16 centimeters long. When we bisect this segment at point \( O \), we divide it into two equal segments:
\[ AO = BO = \frac{AB}{2} = \frac{16 \text{ cm}}{2} = 8 \text{ cm} \]
Next, we bisect both segments \( AO \) and \( BO \) at their midpoints. Let's denote the midpoint of \( AO \) as \( M \) and the midpoint of \( BO \) as \( N \).
Now we calculate the lengths of the new segments \( AM \), \( MO \), \( BN \), and \( NO \):
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For segment \( AO \): \[ AM = MO = \frac{AO}{2} = \frac{8 \text{ cm}}{2} = 4 \text{ cm} \]
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For segment \( BO \): \[ BN = NO = \frac{BO}{2} = \frac{8 \text{ cm}}{2} = 4 \text{ cm} \]
Thus, each of the new segments \( AM \), \( MO \), \( BN \), and \( NO \) is 4 centimeters long.
In summary, each new segment resulting from the bisection of segment \( AO \) and segment \( BO \) is:
\[ \text{Each new segment is } 4 \text{ centimeters long.} \]