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Giải bài 4.23 trang 49 sách bài tập toán 9 - Kết nối tri thức tập 1

Cho A, B là hai địa điểm ở hai bên bờ sông, biết AN và PM cùng vuông góc MN, (MN = n) (mét), (MP = p) (mét), (p > n) và (widehat {MPA} = alpha ) (H.4.12). Chứng minh rằng: (AB = frac{{ptan alpha - n}}{{sin alpha }}).

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

Toán - Văn - Anh
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Đề bài

Cho A, B là hai địa điểm ở hai bên bờ sông, biết AN và PM cùng vuông góc MN, \(MN = n\) (mét), \(MP = p\) (mét), \(p > n\) và \(\widehat {MPA} = \alpha \) (H.4.12). Chứng minh rằng: \(AB = \frac{{p\tan \alpha  - n}}{{\sin \alpha }}\).

Phương pháp giải - Xem chi tiết

+ Chứng minh \(\widehat {BAN} = \widehat {BPM} = \alpha \). + Tam giác BAN vuông tại N có: \(BN = AB.\sin \alpha \). + Tam giác BPM vuông tại M có: \(BM = PM\tan \alpha  = p\tan \alpha \). + \(BM - BN = MN = n\) nên \(p\tan \alpha  - AB\sin \alpha  = n\), từ đó tính được AB theo n, p, \(\alpha \).

Lời giải chi tiết

Vì AN//PM nên \(\widehat {BAN} = \widehat {BPM} = \alpha \). Tam giác BAN vuông tại N có: \(BN = AB.\sin \alpha \). Tam giác BPM vuông tại M có: \(BM = PM\tan \alpha  = p\tan \alpha \). Vì \(BM - BN = MN = n\) nên \(p\tan \alpha  - AB\sin \alpha  = n\). Suy ra \(AB = \frac{{p\tan \alpha  - n}}{{\sin \alpha }}\).

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

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