Final answer:
China's empire was weakened through the colonization of Hong Kong by the British and occupation of Taiwan by Japan, reflecting the era's European colonialism and Japanese imperialism.
Step-by-step explanation:
China's empire was weakened when Hong Kong was colonized by the British and Taiwan was occupied by Japan. These events were significant in the overall decline of China during the period of European colonialism and Japanese imperialism. The colonization of Hong Kong and the occupation of Taiwan are historical facts representing European and Japanese influence over China in the 19th and early 20th centuries.