Is the meaning of the verbs “put out and reach out” the same in this example?
He put out (reached out) an arm and steadied her as she swayed.

Question

Is the meaning of the verbs “put out and reach out” the same  in this example?
He put out (reached out) an arm and steadied her as she swayed.

Answer

No. You would put out your arm to support something, reach it out to grab or touch something. 
He reached out his arm to take the baton.
As the celebrity walked through the crowd, people reached out (their arms) to touch him.

Answer

But be careful with "put out". It has a US English  slang meaning.

Answer

In this context they mean the same thing

Answer

Thank you.

Answer

Yes, in this example they mean the same thing.