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Giải bài tập 2.26 trang 73 SGK Toán 12 tập 1 - Kết nối tri thức

Cho hình hộp ABCD.A’B’C’D’. Lấy M là trung điểm của đoạn thẳng CC’. Vectơ \(\overrightarrow {AM} \) bằng A. \(\overrightarrow {AB} + \overrightarrow {AD} + \overrightarrow {AA'} \). B. \(\overrightarrow {AB} + \overrightarrow {AD} + \frac{1}{2}\overrightarrow {AA'} \). C. \(\overrightarrow {AB} + \frac{1}{2}\overrightarrow {AD} + \frac{1}{2}\overrightarrow {AA'} \). D. \(\frac{1}{2}\overrightarrow {AB} + \overrightarrow {AD} + \overrightarrow {AA'} \).

Tổng ꦗhợp đ♒ề thi học kì 2 lớp 12 tất cả các môn - Kết nối tri thức

Toán - Văn - Anh - Hoá - Sinh - Sử - Địa
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Đề bài

Cho hình hộp ABCD.A’B’C’D’. Lấy M là trung điểm của đoạn thẳng CC’. Vectơ \(\overrightarrow {AM} \) bằng
A. \(\overrightarrow {AB} + \overrightarrow {AD} + \overrightarrow {AA'} \).
B. \(\overrightarrow {AB} + \overrightarrow {AD} + \frac{1}{2}\overrightarrow {AA'} \).
C. \(\overrightarrow {AB} + \frac{1}{2}\overrightarrow {AD} + \frac{1}{2}\overrightarrow {AA'} \).
D. \(\frac{1}{2}\overrightarrow {AB} + \overrightarrow {AD} + \overrightarrow {💫AA'} \).

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về trung điểm của đoạn thẳng để chứng minh: Nếu I là trung điểm của đoạn thẳng AB, với điểm M tùy ý ta có: \(\overrightarrow {MA}  + \overrightarrow {MB}  = 2\overrightarrow {MI} \). Sử dụng kiến thức về quy tắc hình hộp để chứng minh: Cho hình hộp ABCD.A’B’C’D’. Khi đó, ta có: \(\overrightarrow {AB}  + \overrightarrow {AD}  + \overrightarrow {AA'}  = \overrightarrow {AC'} \) Sử dụng quy tắc hình bình hành để chứng minh: Nếu ABCD là hình bình hành thì \(\overrightarrow {AB}  + \overrightarrow {AD}  = \overrightarrow {AC} \)

Lời giải chi tiết

Vì M là trung điểm của CC’ nên \(\overrightarrow {AM}  = \frac{1}{2}\left( {\overrightarrow {AC'}  + \overrightarrow {AC} } \right) = \frac{1}{2}\left( {\overrightarrow {AA'}  + \overrightarrow {AB}  + \overrightarrow {AD}  + \overrightarrow {AB}  + \overrightarrow {AD} } \right)\) \( = \frac{1}{2}\left( {\overrightarrow {AA'}  + 2\overrightarrow {AB}  + 2\overrightarrow {AD} } \right) = \frac{1}{2}\overrightarrow {AA'}  + \overrightarrow {AB}  + \overrightarrow {AD} \) Chọn B.

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