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Giải bài tập 4 trang 79 SGK Toán 9 tập 2 - Cánh diều

Cho tứ giác nội tiếp ABCD có hai tia CD và BA cắt nhau tại I. Chứng minh: a)(widehat {IAD} = widehat {BCD}.) b) IA.IB = ID.IC.

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

Toán - Văn - Anh
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Đề bài

Cho tứ giác nội tiếp ABCD có hai tia CD và BA cắt nhau tại I. Chứng minh: a)\(\widehat {IAD} = \widehat {BCD}.\) b) IA.IB = ID.IC.

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

a) Chứng minh \(\widehat {IAD},\widehat {BCD}\) cùng bù với góc DAB.

b) Chứng minh \(\Delta IAD\backsim \Delta ICB\)(g.g).

Lời giải chi tiết

a) Do tứ giác ABCD nội tiếp đường tròn nên \(\widehat {DAB} + \widehat {DCB} = 180^\circ \). Mà \(\widehat {DAB} + \widehat {IAD} = 180^\circ \) (kề bù) Suy ra \(\widehat {DCB} = \widehat {IAD}\) hay \(\widehat {IAD} = \widehat {BCD}.\) b) Xét tam giác IAD và tam giác ICB có: \(\widehat I\) chung \(\widehat {IAD} = \widehat {BCD}\) (cmt) Nên \(\Delta IAD\backsim \Delta ICB\)(g.g)

Suy ra \(\frac{{IA}}{{ID}} = \frac{{IC}}{{IB}}\) hay IA.IB = IC.ID (đpcm).

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