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Hi, I'm wondering whether it is correct to say about a bicycle or a sledge this way:

1. I rode down the(a?) mountain(the road, etc.) on my (the?) bike.
2. I came down the mountain(the road, etc.) on my bike.
3. I went down the mountain(the roda, etc) on my bike.

Which verb to use?
ride, come or go down?

Is the structure correct?

The same about the sledge

1. I rode down the(a?) mountain on my (the?) sledge.
etc...

By the way, how should I say about a car?
I drove (my car?) down the mountain?

Thanks.
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All your permutations look OK.

"ride" is used for many different modes of transport (bike, horse, bus, train, car etc.). "come" and "go" are generic and needn't refer to modes of transport (you could be walking or running, for instance).

Note that "down the mountain" means literally down, while "down the road" may just mean "along the road".
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Are both correct?

I drove my car down/up the hill.

I drove down the hill on my car.

I rode my bike down/up the hill.

I rode down the hill on my bike.

Should I say "on my bike" or "on the bike"?

down/up "the" or "a" hill?

Thanks.
I drove my car down/up the hill. -- OK

I drove down the hill in my car.

I rode my bike down/up the hill. -- OK

I rode down the hill on my bike. -- OK

"on my bike" -- You're pointing out that the bike belongs to you.

"on the bike" -- It's a specific bike that you've mentioned or is otherwise known to the listener.

"on a bike" -- It's some unspecified bike.

"down/up the hill" -- It's a specific hill that you've mentioned or is otherwise known to the listener.

"down/up a hill" -- It's some unspecified hill.