Final answer:
The most important property lost when a thermometer is broken is its ability to measure temperature.
Step-by-step explanation:
When a thermometer is broken, its ability to measure temperature is compromised. The liquid inside the thermometer, such as mercury or alcohol, which is typically dyed red for visibility, expands and contracts in response to temperature changes. If the thermometer is broken, the most critical property that is lost is the capacity to reliably indicate the temperature because the sealed system that contains the liquid and facilitates its expansion and contraction is disrupted. Thus, the correct answer is C: The temperature can no longer be determined.