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Can I say,

(a) She uses a knife to cut the oranges into / to /in half.

(b) She cut the oranges into / in half last night.
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No, not quite.

You cut something in half (the action).

You cut something into halves (emphasises the pieces).

'In half' is much more common.
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What do you see a "half orange" as being?
Students: We have free audio pronunciation exercises.
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Can I say,

(a) She cut a half orange / a half of orange yesterday
 BarbaraPA's reply was promoted to an answer.
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I mean to nake the orange half. So,

Can I say,

(a) She cut a half orange / a half of orange yesterday.

(b) She cut in half of orange.

(c) She cut the orange half.

(d) She cut in half of the orange.
She cut the orange in half or She cut the orange into halves. Those are your only choices.