Stretches out or makes longer in time or space. Offers or provides something to someone.
From Latin 'extendere' meaning 'to stretch out', from 'ex-' (out) + 'tendere' (to stretch). The metaphorical uses (extending help, extending deadlines) developed from the physical sense of stretching.
When you 'extend your hand' in greeting or 'extend an invitation', you're using the same word Romans used for physically stretching something. The handshake literally embodies the word's original meaning!
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