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Giải bài 7 trang 33 Chuyên đề học tập Toán 11 Cánh diều

Cho tam giác nhọn ABC có trực tâm H. Xác định ảnh của tam giác ABC qua phép vị tự tâm H tỉ số \(k = \frac{1}{2}\).

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Đề bài

Cho tam giác nhọn ABC có trực tâm H. Xác định ảnh của tam giác ABC qua phép vị tự tâm H tỉ số \(k = \frac{1}{2}\).

Phương pháp giải - Xem chi tiết

♊- Nếu phép vị tự tâm O tỉ số k \(\left( {k \ne 0} \right)\) lần lượt biến 2 điểm A, B thành 2 điểm A’, B’ thì \(A'B' = \left| k \right|AB\)

𝄹- Để xác định ảnh của tam giác ABC qua phép vị tự tâm H tỉ số \(k = \frac{1}{2}\), ta xác định ảnh của từng điểm A, B, C qua phép vị tự tâm H tỉ số \(k = \frac{1}{2}\).

Lời giải chi tiết

Gọi A', B', C' lần lượt là ảnh của A, B, C qua phép vị tự tâm H tỉ số \(k = \frac{1}{2}\).Khi đó ta có: \(\overrightarrow {HA'}  = \frac{1}{2}\overrightarrow {HA} ;\,\,\overrightarrow {HB'}  = \frac{1}{2}\overrightarrow {HB} ;\,\,\overrightarrow {HC'}  = \frac{1}{2}\overrightarrow {HC} \)Từ đó suy ra A', B', C' lần lượt là trung điểm của AH, BH, CH.Vậy ảnh của tam giác ABC qua phép vị tự tâm H tỉ số \(k = \frac{1}{2}\) là tam giác A'B'C' với A', B', C' lần lượt là trung điểm của AH, BH, CH.   

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