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Giải bài 6.31 trang 65 SGK Toán 8 - Cùng khám phá

Cho tam giác \(ABC\) vuông tại A có \(AH\) là đường cao.
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Đề bài

Cho tam giác \(ABC\) vuông tại A có \(AH\) là đường cao. Chứng minh rằng: a) Tam giác \(ABC\) đồng dạng với tam giác \(HBA\); b) Tam giác \(ABH\) đồng dạng với tam giác \(CAH\) c) Cho \(BH = 4,CH = 9\). Tính độ dài đường cao \(AH\)

Phương pháp giải - Xem chi tiết

Nếu một góc nhọn của tam giác vuông này bằng một góc nhọn của tam giác vuông kia thì hai tam giác vuông đó đồng dạng.

Lời giải chi tiết

a) Xét tam giác \(ABC\) và tam giác \(HBA\), ta có:\(\widehat A = \widehat {BHA} = 90^\circ \)\(\widehat B\) là góc chung=> \(\Delta ABC\)∽\(\Delta HBA\) (góc nhọn-góc vuông)b) Xét tam giác \(ABH\) và tam giác \(CAH\), ta có:\(\widehat A = \widehat {AHC} = 90^\circ \)\(\widehat {BAH} = \widehat {ACH}\) (do \(\Delta ABC\)∽\(\Delta HBA\))=> \(\Delta ABH\)∽\(\Delta CAH\) (góc nhọn-góc vuông)c) Vì \(\Delta ABH\)∽\(\Delta CAH\), ta có tỉ lệ:\(\begin{array}{l}\frac{{BH}}{{AH}} = \frac{{AH}}{{HC}}\\ \Leftrightarrow A{H^2} = BH.HC\\ \Rightarrow A{H^2} = 4.9 = 36\\ \Rightarrow AH = 6\end{array}\)

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