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Giải bài 2 trang 35 Chuyên đề học tập Toán 11 Chân trời sáng tạo

Các khẳng định sau đúng hay sai?

Tổng ꧑hợp đề thi học kì 2 lớp 1🅰1 tất cả các môn - Chân trời sáng tạo

Toán - Văn - Anh - Lí - Hóa - Sinh
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Đề bài

Các khẳng định sau đúng hay sai? a) Phép vị tự luôn có điểm bất động. b) Phép vị tự không thể có quá một điểm bất động. c) Nếu phép vị tự có hai điểm bất động phân biệt thì mọi điểm đều bất động.

Phương pháp giải - Xem chi tiết

Cho điểm O cố định và một số thực k, \(k \ne 0\). Phép biến hình biến mỗi điểm M thành điểm M’ sao cho \(\overrightarrow {OM'}  = k\overrightarrow {OM} \) được gọi là phép vị tự tâm O tỉ số k, kí hiệu \({V_{(O,k)}}\). O được gọi là tâm vị tự, k gọi là tỉ số vị tự.

Lời giải chi tiết

a) Đúng, vì tâm vị tự là điểm bất động.b) Sai, vì phép vị tự tỉ số \(k{\rm{ }} = {\rm{ }}1\) có mọi điểm đều bất động.c) Đúng.Ta có phép vị tự tâm O luôn có O là điểm bất động.Giả sử phép vị tự đó còn một điểm bất động khác là M.Tức là, ảnh M’ của M qua phép vị tự tâm O trùng với M.Do M’ là ảnh của M qua phép vị tự tâm O, tỉ số k nên \(\overrightarrow {O{M'}}  = k\overrightarrow {OM} \,(1)\)Do M’ trùng với M nên \(\overrightarrow {OM'}  = \overrightarrow {OM} \,\,(2)\)Từ (1), (2), ta suy ra \(k{\rm{ }} = {\rm{ }}1.\)Vì vậy phép vị tự đó là phép đồng nhất.Vậy phép vị tự có hai điểm bất động phân biệt thì mọi điểm đều bất động.

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