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Giải bài 4 trang 94 sách bài tập toán 10 - Chân trời sáng tạo

Cho hình bình hành ABCD có tâm O. Chứng minh rằng:

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Đề bài

Cho hình bình hành ABCD có tâm O. Chứng minh rằng:

a) \(\overrightarrow {CO}  - \overrightarrow {OB}  = \overrightarrow {BA} \)      b) \(\overrightarrow {AB}  - \overrightarrow {BC}  = \overrightarrow {DB} \) c) \(\overrightarrow {DA}  - \overrightarrow {DB}  = \overrightarrow {OD}  - \overrightarrow {OC} \)    d) \(\overrightarrow {DA}  - \overrightarrow {DB}  + \overrightarrow {DC}  = \overrightarrow 0 \)

Phương pháp giải - Xem chi tiết

Sử dụng tính chất của phép cộng, trừ vectơ và quy tắc ba điểm

Lời giải chi tiết

a) Hình bình hành ABCD có tâm O nên \(\overrightarro💟w {CO}  = \overrightarrow {OA} ,\overrightarrow {AB}  = \overrightarrow {DC} ,\overrightarrow {BC}  = \overrightarrow {AD} \)

\(\overrightarrow {CO}  - \overrightarrow {OB}  = \overrightarrow {OA}  - \overrightarrow {OB}  = \overrightarrow {BA} \)        (đpcm)b) \(\overrightarrow {AB}  - \overrightarrow {BC}  = \overrightarrow {DC}  - \overrightarrow {BC}  = \overrightarrow {DC}  + \overrightarrow {CB}  = \overrightarrow {DB} \) (đpcm)c) Ta có:\(\begin{array}{l}\overrightarrow {DA}  - \overrightarrow {DB}  = \overrightarrow {BA} \\\overrightarrow {OD}  - \overrightarrow {OC}  = \overrightarrow {CD} \end{array}\)Mặt khác ta có \(\overrightarrow {BA}  = \overrightarrow {CD} \), suy ra \(\overrightarrow {DA}  - \overrightarrow {DB}  = \overrightarrow {OD}  - \overrightarrow {OC} \) (đpcm)d) \(\overrightarrow {DA}  - \overrightarrow {DB}  + \overrightarrow {DC}  = \left( {\overrightarrow {DA}  - \overrightarrow {DB} } \right) + \overrightarrow {DC}  = \overrightarrow {BA}  + \overrightarrow {DC} \)

Mà ta có ABCD là hình bình hành nên \(\overrightarrow {BA} \) và \(\overrightarrowꦇ {DC} \) là hai vectơ đối nhau

\(\overrightarrow {BA}  + \overrightarrow {DC}  = \overrightarrow 0  \Rightarrow \overrightarrow {DA}  - \overrightarrow {DB}  + \overrightarrow {DC}  = \overrightarrow 0 \)       (đpcm)

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