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Giải bài 53 trang 100 SBT toán 10 - Cánh diều

Cho tam giác ABC, kẻ phân giác AD. Đặt AB = c, AC = b. Chứng minh:

Tổng hợp đề thi 📖học kì 2 lớp 10 tất cả các mônღ - Cánh diều

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Đề bài

Cho tam giác ABC, kẻ phân giác AD. Đặt AB = c, AC = b. Chứng minh:

\(b\overrightarrow {DB}  + c\overrightarrow {DC}  = \overrightarrow 0 \) (*)

Phương pháp giải - Xem chi tiết

Bước 1: Biểu diễn độ dài DB (hoặc DC) theo DC (hoặc DB) và xác định hướng các vectơ tương ứng

Bước 2: Sử dụng định lí đường phân giác trong tam giác để biến đổi tỉ số độ dài \(\frac{{DB}}{{DC}}\) Bước 3: Biến đổi đẳng thức ở bước 1 rồi kết luận

Lời giải chi tiết

Ta có: \(DB = \frac{{DB}}{{DC}}.DC\) mà \(\overrightarrow {DB} \) và \(\overrightarrow {DC} \) ngược hướng  \( \Rightarrow \overrightarrow {DB}  =  - \frac{{DB}}{{DC}}.\overrightarrow {DC} \)(1)

Theo giả thiết, AD là đường phân giác của ∆ABC

\( \Rightarrow \frac{{AB}}{{AC}} = \frac{{DB}}{{DC}} = \frac{c}{b}\) (2) Từ (1) và (2) suy ra \(\overrightarrow {DB}  =  - \frac{c}{b}.\overrightarrow {DC}  \Leftrightarrow b\overrightarrow {DB}  =  - c\overrightarrow {DC}  \Leftrightarrow b\overrightarrow {DB}  + c\overrightarrow {DC}  = \overrightarrow 0 \) (ĐPCM)

 

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