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Giải bài 42 trang 76 sách bài tập toán 8 – Cánh diều

Cho tam giác (ABC) vuông ở (A) có (AB = 3AC) và điểm (D) thuộc cạnh (AB) sao cho (AD = 2DB). Chứng minh: (widehat {ADC} + widehat {ABC} = 45^circ ).

🗹Tổng hợp đề thi học kì 2 lớp 8 tất cả các môn - Cánh diều

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

Cho tam giác \(ABC\) vuông ở \(A\) có \(AB = 3AC\) và điểm \(D\) thuộc cạnh \(AB\) sao cho \(AD = 2DB\). Chứng minh: \(\widehat {ADC} + \widehat {ABC} = 45^\circ \).

Phương pháp giải - Xem chi tiết

Áp dụng trường hợp đồng dạng thứ hai của tam giác: cạnh – góc – cạnh Nếu hai cạnh của tam giác này tỉ lệ với hai cạnh của tam giác kia và hai góc tạo bởi các cặp cạnh đó bằng nhau thì hai tam giác đó đồng dạng.

Lời giải chi tiết

Gọi \(E\) là trung điểm của \(AD\). Đặt \(AE = x,AC = x\). Có \(AE = ED = DB,AB = 3AC\) nên \(ED = x,EB = 2x\) và \(CE = x\sqrt 2 \). Xét tam giác \(EDC\) và tam giác \(ECB\) có: \(\widehat {CED} = \widehat {CEB}\) \(\frac{{ED}}{{EC}} = \frac{{EC}}{{EB}}\) nên \(\Delta EDC\backsim \Delta ECB\) (c.g.c). Do đó \(\widehat {ECD} = \widehat {CEB}\). Vì vậy \(\widehat {ADC} + \widehat {ABC} = \widehat {EDC} + \widehat {ECD} = \widehat {AEC}\). Mặt khác, do tam giác \(AEC\) là tam giác vuông cân nên \(\widehat {AEC} = 45^\circ \). Vậy \(\widehat {ADC} + \widehat {ABC} = 45^\circ \).

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