A city acquires a 10-foot strip of a property owner's land for a public walking trail. the owner's home is not affected and she is able to use the land. is the homeowner eligible for compensation?

In this situation, yes, the homeowner eligible for compensation

the government has a power called power of Eminent Domain, which is a power to convert privately-owned property into a public infrastructure in exchange of compensation, as long as the property owner is not directly affected by the new infrastructure