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Giải bài 6.30 trang 65 SGK Toán 8 - Cùng khám phá

Cho tam giác \(ABC\) có hai đường cao
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Đề bài

Cho tam giác \(ABC\) có hai đường cao $BD$ và $CE$ cắt nhau tại O. Chứng minh rẳng: a) Tam giác \(ABD\) đồng dạng với tam giác b) \(OE.OC = OB.OD\)\(ACE\)

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 \(ABD\) và tam giác \(ACE\), ta có:\(\widehat A\) là góc chung\(\widehat {ADB} = \widehat {AEC} = 90^\circ \)=> \(\Delta ABD\)∽\(\Delta ACE\) (góc vuông-góc nhọn)b) Xét tam giác \(OEB\) và tam giác \(ODC\), ta có:\(\widehat {OEB} = \widehat {ODC} = 90^\circ \)\(\widehat {EOB} = \widehat {DOC}\) (2 góc đối đỉnh)=> \(\Delta OEB\)∽\(\Delta ODC\) (góc vuông-góc nhọn)Ta có tỉ lệ đồng dạng:\(\begin{array}{l}\frac{{OE}}{{OD}} = \frac{{OB}}{{OC}}\\ \Leftrightarrow OE.OC = OB.OD(dpcm)\end{array}\)

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