Cho hình vuông tâm \(O\). Hỏi có bao nhiêu phép quay tâm \(O\) góc \(\alpha \), \(0 \le \alpha < 2\pi \), biến hình vuông trên thành chính nó?

Ta có:

+) \({Q_{\left( {O,0} \right)}}\left( A \right) = A,\) \({Q_{\left( {O,0} \right)}}\left( B \right) = B,\) \({Q_{\left( {O,0} \right)}}\left( C \right) = C,\) \({Q_{\left( {O,0} \right)}}\left( D \right) = D\)

Do đó \({Q_{\left( {O,0} \right)}}\left( {ABCD} \right) = ABCD\).

+) \({Q_{\left( {O,\dfrac{\pi }{2}} \right)}}\left( A \right) = B,\) \({Q_{\left( {O,\dfrac{\pi }{2}} \right)}}\left( B \right) = C,\) \({Q_{\left( {O,\dfrac{\pi }{2}} \right)}}\left( C \right) = D,\) \({Q_{\left( {O,\dfrac{\pi }{2}} \right)}}\left( D \right) = A\)

Do đó \({Q_{\left( {O,\dfrac{\pi }{2}} \right)}}\left( {ABCD} \right) = BCDA\).

+) \({Q_{\left( {O,\pi } \right)}}\left( A \right) = C,\) \({Q_{\left( {O,\pi } \right)}}\left( B \right) = D,\) \({Q_{\left( {O,\pi } \right)}}\left( C \right) = A,\) \({Q_{\left( {O,\pi } \right)}}\left( D \right) = B\)

Do đó \({Q_{\left( {O,\dfrac{\pi }{2}} \right)}}\left( {ABCD} \right) = CDAB\).

+) \({Q_{\left( {O,\dfrac{{3\pi }}{2}} \right)}}\left( A \right) = D,\) \({Q_{\left( {O,\dfrac{{3\pi }}{2}} \right)}}\left( B \right) = A,\) \({Q_{\left( {O,\dfrac{{3\pi }}{2}} \right)}}\left( C \right) = B,\) \({Q_{\left( {O,\dfrac{{3\pi }}{2}} \right)}}\left( D \right) = C\)

Do đó \({Q_{\left( {O,\dfrac{\pi }{2}} \right)}}\left( {ABCD} \right) = DABC\).

Vậy có \(4\) phép quay cần tìm.

Chọn D.