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For every position that the hour hand CAN attain, there are TWO points where the minute hand will achieve a ninety degree angle from the hour hand.

Answer

It happens 44 times in the course of one full day, and it's reasonably easy to calculate. See linked question for details.

Technically, if you move the hands of the clock by hand, an infiinite number of right angles can be formed.

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Q: How many times do the clock hands form a 90 degree angle?
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