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Giải bài 5 trang 76 SGK Toán 8 tập 2– Chân trời sáng tạo

Quan sát Hình 12. Chứng minh rằng:

Tổng hợp đề thi học kì 2 lớp 8 tất c꧙ả các môn - Chân tr💯ời sáng tạo

Toán - Văn - Anh - Khoa học tự nhiên
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Đề bài

Quan sát Hình 12. Chứng minh rằng:

 

a) \(\Delta ABH\backsim\Delta DCB\); b) \(\frac{{BC}}{{BE}} = \frac{{BD}}{{BA}}\).

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

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

Lời giải chi tiết

a) Vì \(\left\{ \begin{array}{l}JC \bot AE\\BH \bot AE\end{array} \right. \) nên \(JC//BH\). Vì \(JC//BH \) nên \(\widehat {HBA} = \widehat {JCA}\) (hai góc đồng vị) hay \(\widehat {HBA} = \widehat {DCB}\) Xét \(\Delta ABH\) và \(\Delta DCB\) có: \(\widehat {HBA} = \widehat {DCB}\) (chứng minh trên) \(\widehat {AHB} = \widehat {DBC} = 90^\circ \) Do đó, \(\Delta ABH\backsim\Delta DCB\) (g.g) b) Vì  (hai góc tương ứng) hay \(\widehat {EAB} = \widehat {CDB}\). Xét \(\Delta AEB\) và \(\Delta DCB\) có: \(\widehat {EAB} = \widehat {CDB}\) (chứng minh trên) \(\widehat {ABE} = \widehat {DBC} = 90^\circ \) Do đó, \(\Delta AEB\backsim\Delta DCB\) (g.g) Suy ra, \(\frac{{BE}}{{BC}} = \frac{{BA}}{{BD}}\) (các cặp cạnh tương ứng có cùng tỉ lệ) Hay \(\frac{{BC}}{{BE}} = \frac{{BD}}{{BA}}\) (điều phải chứng minh).

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