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Giải bài 30 trang 73 SBT toán 10 - Cánh diều

Cho tam giác ABC có A(3 ; 7), B(–2 ; 2), C(6 ; 1). Viết phương trình tổng quát của các đường cao của tam giác ABC.

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Đề bài

Cho tam giác ABC có A(3 ; 7), B(–2 ; 2), C(6 ; 1). Viết phương trình tổng quát của các đường cao của tam giác ABC.

Phương pháp giải - Xem chi tiết

Bước 1: Tìm tọa độ VTPT của các đường cao là cạnh đối diện tương ứng Bước 2: Tìm điểm đi qua là các đỉnh của tam giác Bước 3: Viết PTTQ của các đường cao khi biết điểm đi qua và VTPT tương ứng

Lời giải chi tiết

Ta có: \(\overrightarrow {AB}  = ( - 5; - 5),\overrightarrow {AC}  = (3; - 6),\overrightarrow {BC}  = (8; - 1)\)

Gọi AH, BM, CN là các đường cao của ∆ABC. Khi đó:

+ \(AH \bot BC \Rightarrow \) AH đi qua A và nhận \(\overrightarrow {BC}  = (8; - 1)\) làm VTPT nên có PT: 8x – y – 17 = 0

+ \(BM \bot AC \Rightarrow \) BM đi qua B và nhận \(\overrightarrow𝔉 {{n_1}}  = (1; - 2)\) cùng phương với \(\overrightarrow {AC}  = (3; - 6)\) làm VTPT nên có PT:

x – 2y + 6 = 0

+ \(CN \bot AB \Rightarrow \) CN đi qua C và nhận \(\overrightarrow {{n_2}}  = (⛦1;1)\) cùng phương với \(\overrightarrow {AB}  = ( - 5; - 5)\⛎) làm VTPT nên có PT:

x + y – 7 = 0

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