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Giải bài 9.51 trang 62 sách bài tập toán 9 - Kết nối tri thức tập 2

Cho tam giác nhọn ABC có đường cao AH. Các điểm M và N lần lượt là hình chiếu vuông góc của H trên AB, AC. Chứng minh rằng (AM.AB = AN.AC).

Tổng hợp Đề thi vào 10 có đáp án và lời giải

Toán - Văn - Anh
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Đề bài

Cho tam giác nhọn ABC có đường cao AH. Các điểm M và N lần lượt là hình chiếu vuông góc của H trên AB, AC. Chứng minh rằng \(AM.AB = AN.AC\).

Phương pháp giải - Xem chi tiết

+ Chứng minh tứ giác AMHN nội tiếp đường tròn đường kính AH, suy ra \(\widehat {AMN} = \widehat {AHN}\). + Ta có: \(\widehat {AMN} = \widehat {AHN} = {90^o} - \widehat {HAN} = \widehat {ACB}\). + Chứng minh $\Delta AMN\backsim \Delta ACB\left( g.g \right)$, suy ra \(AM.AB = AN.AC\).

Lời giải chi tiết

Vì \(\widehat {AMH} = \widehat {ANH} = {90^o}\) nên tam giác AMH vuông tại M và tam giác ANH vuông tại N. Suy ra, hai tam giác AMH, ANH nội tiếp đường tròn đường kính AH. Suy ra, tứ giác AMHN nội tiếp đường tròn đường kính AH. Do đó, \(\widehat {AMN} = \widehat {AHN}\) (hai góc nội tiếp đường tròn đường kính AH cùng chắn cung AN). Ta có: \(\widehat {AMN} = \widehat {AHN} = {90^o} - \widehat {HAN} = \widehat {ACB}\) Tam giác AMN và tam giác ACB có: \(\widehat {AMN} = \widehat {ACB}\) (cmt), \(\widehat {MAN} = \widehat {CAB}\) (góc chung) nên $\Delta AMN\backsim \Delta ACB\left( g.g \right)$. Suy ra \(\frac{{AM}}{{AC}} = \frac{{AN}}{{AB}}\), suy ra \(AM.AB = AN.AC\).

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Tham Gia Group Dành Cho Lớp 9 - Ôn Thi Vào Lớp 10 Miễn Phí

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