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Giải bài tập 9.19 trang 83 SGK Toán 9 tập 2 - Kết nối tri thức

Cho điểm I nằm ngoài đường tròn (O). Qua I kẻ hai đường thẳng lần lượt cắt (O) tại bốn điểm A, B và C, D sao cho A nằm giữa B và I, C nằm giữa D và I. Chứng minh rằng (widehat {IBD} = widehat {ICA},widehat {IAC} = widehat {IDB}) và (IA.IB = IC.ID).

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

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Đề bài

Cho điểm I nằm ngoài đường tròn (O). Qua I kẻ hai đường thẳng lần lượt cắt (O) tại bốn điểm A, B và C, D sao cho A nằm giữa B và I, C nằm giữa D và I. Chứng minh rằng \(\widehat {IBD} = \widehat {ICA},\widehat {IAC} = \widehat {IDB}\) và \(IA.IB = IC.ID\).

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

+ Chứng minh \(\widehat {ABD} + \widehat {ACD} = {180^o}\), mà \(\widehat {ICA} + \widehat {ACD} = {180^o}\) nên \(\widehat {IBD} = \widehat {ICA}\). + Chứng minh \(\widehat {CAB} + \widehat {CDB} = {180^o}\), mà \(\widehat {CAB} + \widehat {IAC} = {180^o}\) nên \(\widehat {IAC} = \widehat {IDB}\). + Chứng minh $\Delta IAC\backsim \Delta IDB\Rightarrow \frac{IA}{IC}=\frac{ID}{IB}\Rightarrow IA.IB=IC.ID$.

Lời giải chi tiết

Tứ giác ABDC nội tiếp (O) nên \(\widehat {ABD} + \widehat {ACD} = {180^o}\), mà \(\widehat {ICA} + \widehat {ACD} = {180^o}\) (hai góc kề bù) nên \(\widehat {IBD} = \widehat {ICA}\) Tứ giác ABDC nội tiếp (O) nên\(\widehat {CAB} + \widehat {CDB} = {180^o}\), mà \(\widehat {CAB} + \widehat {IAC} = {180^o}\) (hai góc kề bù) nên \(\widehat {IAC} = \widehat {IDB}\) Tam giác IAC và tam giác IDB có: Góc I chung  \(\widehat {ICA} = \widehat {IBD}\) (cmt). Do đó, $\Delta IAC\backsim \Delta IDB$ nên $\frac{IA}{IC}=\frac{ID}{IB}$ suy ra $IA.IB=IC.ID$.

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

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