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Giải bài 30 trang 115 sách bài tập toán 9 - Cánh diều tập 2

Trên mặt phẳng tọa độ Oxy cho hình vuông ABCD với A(0; 2), B(–2; 0), C(0; –2), D(2; 0). Phép quay thuận chiều 90° tâm O biến các điểm A, B, C, D lần lượt thành các điểm A’, B’, C’, D’. Tính chu vi tứ giác A’B’C’D’.

Tổng hợp Đề thi vào 10 có đáp án và lời giải

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Đề bài

💝Trên mặt phẳng tọa độ Oxy cho hình vuông ABCD với A(0; 2), B(–2; 0), C(0; –2), D(2; 0). Phép quay thuận chiều 90° tâm O biến các điểm A, B, C, D lần lượt thành các điểm A’, B’, C’, D’. Tính chu vi tứ giác A’B’C’D’.

Phương pháp giải - Xem chi tiết

ꦓDựa vào phép quay thuận chiều \({\alpha ^o}\) (\({0^o} < {\alpha ^o} < {360^o}\)) tâm O giữ nguyên điểm O, biến điểm M (khác điểm O) thành điểm M’ thuộc đường tròn (O; OM) sao cho tia OM quay thuận chiều kim đồng hồ đến tia OM’ thì điểm M tạo nên cung MnM’ có số đo \({\alpha ^o}\).

♚Dựa vào phép quay thuận chiều \({\alpha ^o}\) (\({0^o} < {\alpha ^o} < {360^o}\)) tâm O được phát biểu tương tự như trên.

Lời giải chi tiết

Do ABCD là hình vuông nên hai đường chéo AC và BD vuông góc với nhau tại trung điểm của mỗi đường. Ta có A(0; 2), B(–2; 0), C(0; –2), D(2; 0) nên B, D cùng nằm trên Ox và A, C cùng nằm trên Oy. Ta cũng suy ra được OA = OB = OC = OD hay O là tâm của hình vuông ABCD. Xét ∆OAB vuông tại O, theo định lí Pythagore, ta có:

AB2 = OA2 + OB2 = 22 + 22 = 8.

Suy ra \(AB = \sqrt 8  = 2\sqrt 2 \). Như vậy, hình vuông ABCD có cạnh bằng \(2\sqrt 2 \). Ta có phép quay thuận chiều 90° tâm O giữ nguyên hình vuông ABCD do đó chu vi tứ giác A’B’C’D’ bằng chu vi hình vuông ABCD và bằng 4. \(2\sqrt 2  = 8\sqrt 2 \) (đơn vị chiều dài).

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