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Giải bài 6 trang 98 SGK Toán 10 tập 1 – Cánh diều

Cho tam giác nhọn ABC, kẻ đường cao AH. Chứng minh rằng:

ඣTổng hợp đề thi học kì 2 lớp 10 tất cả các môn - Cánh diều

Toán - Văn - Anh - Lí - Hóa - Sinh - Sử - Địa...
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Đề bài

Cho tam giác nhọn ABC, kẻ đường cao AH. Chứng minh rằng: a) \(\overrightarrow {AB} .\overrightarrow {AH}  = \overrightarrow {AC} .\overrightarrow {AH} \) b) \(\overrightarrow {AB} .\overrightarrow {BC}  = \overrightarrow {HB} .\overrightarrow {BC} \)

Phương pháp giải - Xem chi tiết

Áp dụng tính chất phân phối:  \(\overrightarrow {AB} .\overrightarrow {AH}  - \overrightarrow {AC} .\overrightarrow {AH}  = (\overrightarrow {AB}  - \overrightarrow {AC} ).\overrightarrow {AH} \) Quy tắc hiệu: \(\overrightarrow {AB}  - \overrightarrow {AC}  = \overrightarrow {BC} \)

Lời giải chi tiết

Ta có: \(AH \bot CB \Rightarrow (\overrightarrow {AH} ,\overrightarrow {CB} ) = {90^o} \Leftrightarrow \cos (\overrightarrow {AH} ,\overrightarrow {CB} ) = 0 \Leftrightarrow \overrightarrow {AH} .\overrightarrow {CB}  = 0\)a) \(\overrightarrow {AB} .\overrightarrow {AH}  - \overrightarrow {AC} .\overrightarrow {AH}  = (\overrightarrow {AB}  - \overrightarrow {AC} ).\overrightarrow {AH}  = \overrightarrow {CB} .\overrightarrow {AH}  = 0\)\( \Leftrightarrow \overrightarrow {AB} .\overrightarrow {AH}  = \overrightarrow {AC} .\overrightarrow {AH} \)b)  \(\overrightarrow {AB} .\overrightarrow {BC}  - \overrightarrow {HB} .\overrightarrow {BC}  = (\overrightarrow {AB}  - \overrightarrow {HB} ).\overrightarrow {BC}  = (\overrightarrow {AB}  + \overrightarrow {BH} ).\overrightarrow {BC}  = \overrightarrow {AH} .\overrightarrow {BC}  = 0\)\( \Leftrightarrow \overrightarrow {AB} .\overrightarrow {BC}  = \overrightarrow {HB} .\overrightarrow {BC} \)

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