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Giải bài 17 trang 101 sách bài tập toán 9 - Chân trời sáng tạo tập 1

Cho hai đường tròn (O; R) và (O’; R’) cắt nhau tại hai điểm A và B phân biệt. Vẽ đường thẳng vuông góc với AB tại A và cắt (O), (O’) lần lượt tại C, D. Tia CB cắt (O’) tại E, tia DB cắt (O) tại F. Chứng minh rằng: a) CD.CA = CB.CE. b) DC.DA = DB.DF. c) CD2 = CB.CE + DB.DF.

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

Toán - Văn - Anh
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Đề bài

Cho hai đường tròn (O; R) và (O’; R’) cắt nhau tại hai điểm A và B phân biệt. Vẽ đường thẳng vuông góc với AB tại A và cắt (O), (O’) lần lượt tại C, D. Tia CB cắt (O’) ꦛtại E, tia DB cắt (O) tại F. Chứng minh rằng:

a) CD.CA = CB.CE.

b) DC.DA = DB.DF.

c) CD2 = CB.CE + DB.DF.

Phương pháp giải - Xem chi tiết

Dựa vào: Hai góc nội tiếp cùng chắn một cung thì bằng nhau

Góc nội tiếp bằng nửa số đo cung bị chắn.

Lời giải chi tiết

a) Xét tam giác CDB và tam giác CEA có góc C chung. Trong đường tròn (O’), ta có: \(\widehat{CDB}=\widehat{ADB}=\frac{1}{2}sđ\overset\frown{AB}\),  \(\widehat{CEA}=\widehat{BEA}=\frac{1}{2}sđ\overset\frown{AB}\) Suy ra \(\widehat {CDB} = \widehat {CEA}\), do đó \(\Delta CDB\backsim \Delta CEA\) Suy ra \(\frac{{CD}}{{CE}} = \frac{{CB}}{{CA}}\) hay CD.CA = CB.CE. b) Chứng minh tương tự, ta cũng có DC.DA = DB.DF. c) Ta có:

CB.CE + DB.DF = CD.CA + DC.DA = CD(CA + AD) = CD.CD = CD2.

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