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Giải bài 9.42 trang 60 sách bài tập toán 9 - Kết nối tri thức tập 2

Một phép quay thuận chiều ({120^o}) tâm O biến điểm A thành điểm B, biến điểm B thành điểm C. Chứng tỏ rằng tam giác ABC là tam giác đều nội tiếp một đường tròn tâm O.

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

Toán - Văn - Anh
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Đề bài

Một phép quay thuận chiều \({120^o}\) tâm O biến điểm A thành điểm B, biến điểm B thành điểm C. Chứng tỏ rằng tam giác ABC là tam giác đều nội tiếp một đường tròn tâm O.

Phương pháp giải - Xem chi tiết

+ Tính được các cung nhỏ AB, BC có số đo bằng \({120^o}\). + Tính được \(\widehat {ACB} = \frac{1}{2}\)sđ$\overset\frown{AB}={{60}^{o}}$, \(\widehat {BAC} = \frac{1}{2}\)sđ$\overset\frown{BC}={{60}^{o}}$, \(\widehat {ABC} = {180^o} - \widehat {ACB} - \widehat {BAC} = {60^o}\) nên tam giác ABC đều.

Lời giải chi tiết

Ta có: \(OA = OB = OC\) nên tam giác ABC nội tiếp (O) và các cung nhỏ AB, BC có số đo bằng \({120^o}\).

Vì góc ACB và góc BAC là các góc nội tiếp đường tròn (O) lần lượt chắn các cung nhỏ AB, BC nên \(\widehat {ACB} = \frac{1}{2}\)sđ$\overset\frown{AB}$nhỏ\( = {60^o}\), \(\widehat {BAC} = \frac{1}{2}\)sđ$\overset\frown{BC}$ nhỏ \( = {60^o}\).

Tam giác ABC có: \(\widehat {ABC} = {180^o} - \widehat {ACB} - \widehat {BAC} = {60^o}\). Tam giác ABC có: \(\widehat {ABC} = \widehat {ACB} = \widehat {CAB} = {60^o}\) nên tam giác ABC đều.

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

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