Collocations are words that usually go together in English.
Here we are going to look at words that go with the word 'hand' to make useful and popular expressions you will hear in social and professional contexts.
If you have a free hand it means one of your hands is empty or not occupied with doing anything. One of your hands could be busy and the other is a free hand. It can also be used in more professional contexts to express that someone is trusting you to use your skills and judgement to make decisions without restrictions (the same as giving someone 'free rein'). Don't confuse this collocation with the word 'freehand', which means to draw, write or sketch something without using any additional tools or help.
To have a sympathetic hand means to behave in a way which is sympathetic or showing understanding towards somebody else. It can be used in both social and professional contexts, but it is mainly used to deal with an emotional situation or when someone is suffering.
When we speak of busy hands we mean hands that are occupied or doing something or focused on a task. It can also be used in a colloquial or slang context in order to describe inappropriate or unwanted physical behaviour or touching.
When you shake hands your right hand meets another right hand in a movement that can be a greeting or an apology. In business situations, you might shake hands in order to cement an agreement. This is an everyday expression that you will hear a lot.
Extend a hand
When you extend a hand it means to hold out your hand in order for someone to shake it.
It can also be used figuratively such as in the phrase 'extend the hand of friendship', in order to talk about making an offer.