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Giải bài tập 9.13 trang 79 SGK Toán 9 tập 2 - Kết nối tri thức

Cho tam giác ABC nội tiếp đường tròn (O). Biết rằng (widehat {BOC} = {120^o}) và (widehat {OCA} = {20^o}). Tính số đo các góc của tam giác ABC.

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

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Đề bài

Cho tam giác ABC nội tiếp đường tròn (O). Biết rằng \(\widehat {BOC} = {120^o}\) và \(\widehat {OCA} = {20^o}\). Tính số đo các góc của tam giác ABC.

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

+ Tính được \(\widehat {BAC} = \frac{1}{2}\widehat {BOC} = \frac{1}{2}{.120^o} = {60^o}\). + Chứng minh tam giác AOC cân tại O, tính được: \(\widehat {ACO} = \widehat {OAC}\) + Tính được \(\widehat {AOC} = {180^o} - \widehat {CAO} - \widehat {ACO}\) + Tính được \(\widehat {ABC} = \frac{1}{2}\widehat {AOC} = \frac{1}{2}{.140^o} = {70^o}\) + Tam giác ABC có: \(\widehat {ACB} = {180^o} - \widehat {BAC} - \widehat {ABC}\)

Lời giải chi tiết

Tam giác ACO có: \(OA = OC\) (bán kính (O)) nên tam giác AOC cân tại O. Do đó, \(\widehat {ACO} = \widehat {OAC} = {20^o}\) Suy ra: \(\widehat {AOC} = {180^o} - \widehat {CAO} - \widehat {ACO} = {180^o} - {20^o} - {20^o} = {140^o}\) Xét đường tròn (O): Vì góc nội tiếp BAC và góc ở tâm BOC cùng chắn cung nhỏ BC nên \(\widehat {BAC} = \frac{1}{2}\widehat {BOC} = \frac{1}{2}{.120^o} = {60^o}\) Vì góc nội tiếp ABC và góc ở tâm AOC cùng chắn cung nhỏ AC nên \(\widehat {ABC} = \frac{1}{2}\widehat {AOC} = \frac{1}{2}{.140^o} = {70^o}\) Tam giác ABC có: \(\widehat {ACB} = {180^o} - \widehat {BAC} - \widehat {ABC} = {180^o} - {60^o} - {70^o} = {50^o}\)

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

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