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Giải Bài 9.9 trang 50 sách bài tập toán 7 - Kết nối tri thức với cuộc sống

Cho tam giác ABC cân tại A và một điểm M tuỳ ý thuộc đoạn thẳng BC. Chứng minh rằng tổng khoảng cách từ điểm M đến đường thẳng AB, AC là một số không đổi

Tổng hợp đề thi học kì 2 lớp 7 tất cả các môꦑn🍬 - Kết nối tri thức

Toán - Văn - Anh - Khoa học tự nhiên...
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Đề bài

Cho tam giác ABC cân tại A và một điểm M tuỳ ý thuộc đoạn thẳng BC. Chứng minh rằng tổng khoảng cách từ điểm M đến đường thẳng AB, AC là một số không đổi

Phương pháp giải - Xem chi tiết

- Xét khi M trùng B, C và khi M khác B, C - Kẻ \(MP \bot AC;MQ \bot AB\) -Chứng minh: \(\Delta RBM\) cân tại R -Chứng minh: MP + MQ = BS + SI = BI = CK.

Lời giải chi tiết

TH1:Khi M trùng với B hay C thì tổng khoảng cách đó là BIℱ hoặc CK

Theo bài 9.8: BI = CK

TH2: Khi M khác B, khác C

Kẻ \(MP \bot AC;MQ \bot AB\)\( \Rightarrow \)Tổng khoảng cách từ điểm M đến đường thẳng AB, AC là: MQ + MPQua M kẻ \(MR // AC\); MR cắt BI tại S.\( \Rightarrow \widehat C = \widehat {RMB}\) (2 góc đồng vị)Mà \(\widehat C = \widehat B\)\( \Rightarrow \widehat B = \widehat {RMB}\)\( \Rightarrow \Delta RBM\) cân tại RMQ là khoảng cách từ M đến RB, BS là khoảng cách từ B đến RMTheo bài 9.8: MQ = BSTa có: MR // AC, MP và SI có độ dài là khoảng cách giữa hai đường thẳng đó nên MP = SISuy ra: MP + MQ = BS + SI = BI = CK. 

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