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Giải bài 1 trang 97 SGK Toán 10 tập 1 – Chân trời sáng tạo

Cho hình bình hành ABCD D có O là giao điểm hai đường chéo. Với M là điểm tùy ý, chứng minh rằng:

Tổng hợp đề 𒅌thi học kì 2 lớp 10 tất cả các môn - Chân trời sáng tạo

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Đề bài

Cho hình bình hành ABCD có O là giao điểm hai đường chéo. Với M là điểm tùy ý, chứng minh rằng:

a) \(\overrightarrow {MA}  + \overrightarrow {MB}  + \overrightarrow {MC}  + \overrightarrow {MD}  = 4\overrightarrow {MO} \)

b) \(\overrightarrow {AB}  + \overrightarrow {AC}  + \overrightarrow {AD}  = 2\overrightarrow {AC} \)

Phương pháp giải - Xem chi tiết

a) Sử dụng quy tắc ba điểm \(\overrightarrow {MA}  = \overrightarrow {MO}  + \overrightarrow {OA} \) và tính chất trung điểm \(\overrightarrow {OA}  + \overrightarrow {OC}  = \overrightarrow 0 \) b) Sử dụng tính chất của bình bình hành \(\overrightarrow {AB}  + \overrightarrow {AD}  = \overrightarrow {AC} \)

Lời giải chi tiết

a) \(\overrightarrow {MA}  + \overrightarrow {MB}  + \overrightarrow {MC}  + \overrightarrow {MD}  = 4\overrightarrow {MO} \)\( \Leftrightarrow \overrightarrow {MO}  + \overrightarrow {OA}  + \overrightarrow {MO}  + \overrightarrow {OB}  + \overrightarrow {MO}  + \overrightarrow {OC}  + \overrightarrow {MO}  + \overrightarrow {OD}  = 4\overrightarrow {MO} \)\( \Leftrightarrow 4\overrightarrow {MO}  + \left( {\overrightarrow {OA}  + \overrightarrow {OB} } \right) + \left( {\overrightarrow {OC}  + \overrightarrow {OD} } \right) = 4\overrightarrow {MO} \)\( \Leftrightarrow 4\overrightarrow {MO}  + \overrightarrow 0  + \overrightarrow 0  = 4\overrightarrow {MO} \\ \Leftrightarrow 4\overrightarrow {MO}  = 4\overrightarrow {MO} \) (luôn đúng)

(vì O là giao điểm 2 đường chéo nên là trung điểm của AB, CD)

b) ABCD là hình ꧂bình hành nên ta có \(\overrightarrow {AB}  + \overrightarrow {AD}  = \overrightarrow {AC} \)

Suy ra \(\)\(\overrightarrow {AB}  + \overrightarrow {AC}  + \overrightarrow {AD} = \left( {\overrightarrow {AB}  + \overrightarrow {AD} } \right) + \overrightarrow {AC}  = \overrightarrow {AC}  + \overrightarrow {AC}  = 2\overrightarrow {AC} \) (đpcm)

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