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Giải bài 4.19 trang 48 sách bài tập toán 9 - Kết nối tri thức tập 1

Gọi AH là đường cao của tam giác ABC vuông tại A. Tính (tan widehat {ABH}) và (tan widehat {CAH}), suy ra (A{H^2} = BH.CH).

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

Toán - Văn - Anh
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Đề bài

Gọi AH là đường cao của tam giác ABC vuông tại A. Tính \(\tan \widehat {ABH}\) và \(\tan \widehat {CAH}\), suy ra \(A{H^2} = BH.CH\).

Phương pháp giải - Xem chi tiết

+ Xét tam giác ABC vuông tại A có góc nhọn B bằng \(\alpha \) thì tỉ số giữa cạnh đối và cạnh kề gọi là tan của \(\alpha \). + Hai góc phụ nhau thì tan góc này bằng côtang góc kia.

Lời giải chi tiết

Tam giác AHC vuông tại H nên \(\tan \widehat {CAH} = \frac{{HC}}{{AH}}\), \(\tan \widehat {ACH} = \frac{{AH}}{{HC}}\) Tam giác AHB vuông tại H nên \(\tan \widehat {ABH} = \frac{{AH}}{{BH}}\). Hai góc ABH và ACH là hai góc phụ nhau nên \(\tan \widehat {ABH} = \cot \widehat {ACH} = \frac{1}{{\tan \widehat {ACH}}}\). Do đó \(\frac{{AH}}{{BH}} = \frac{{HC}}{{AH}}\). Suy ra \(A{H^2} = BH.CH\).

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

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