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Giải bài 4.53 trang 68 sách bài tập toán 10 - Kết nối tri thức với cuộc sống

Cho tam giác ABC có AB = 1,BC = 2

😼Tổng hợp đề thi học kì 2 lớp 10 tất cả các môn - Kết nối tri thức

Toán - Văn - Anh - Lí - Hóa - Sinh - Sử - Địa...
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Đề bài

Cho tam giác \(ABC\) có \(AB = 1,\,\,BC = 2\) và \(\widehat {ABC} = {60^ \circ }.\) Tích vô hướng \(\overrightarrow {BC} .\overrightarrow {CA} \) bằng A. \(\sqrt 3 \) B. \( - \sqrt 3 \) C. \(3\) D. \( - 3\)

Phương pháp giải - Xem chi tiết

-   Áp dụng định lý cosin để tính \(AC\): \(A{C^2} = A{B^2} + B{C^2} - 2AB.BC.\cos \widehat {ABC}\) -   Áp dụng định lý sin để tính góc \(\widehat {ACB}\): \(\frac{{AB}}{{\sin \widehat {ACB}}} = \frac{{AC}}{{\sin \widehat {ABC}}}\) -  Áp dụng công thức tính tích vô hướng của \(\overrightarrow {BC} .\overrightarrow {CA} \)

Lời giải chi tiết

Gọi \(D\) là điểm đối xứng với \(B\) qua \(C\)

Áp dụng định lý cosin, ta có:\(\begin{array}{l}A{C^2} = A{B^2} + B{C^2} - 2AB.BC.\cos \widehat {ABC}\\ \Rightarrow \,\,A{C^2} = 1 + 4 - 2.1.2.\cos {60^ \circ } = 3\\ \Rightarrow \,\,AC = \sqrt 3 \end{array}\)Áp dụng định lý sin, ta có:\(\begin{array}{l}\frac{{AB}}{{\sin \widehat {ACB}}} = \frac{{AC}}{{\sin \widehat {ABC}}}\,\, \Leftrightarrow \,\,\frac{1}{{\sin \overrightarrow {ACB} }} = \frac{{\sqrt 3 }}{{\sin {{60}^ \circ }}}\\ \Leftrightarrow \,\,\sin \widehat {ACB} = \frac{{\sin {{60}^ \circ }}}{{\sqrt 3 }} = \frac{1}{2}\\ \Leftrightarrow \,\,\widehat {ACB} = {30^ \circ }\,\, \Rightarrow \,\,\widehat {ACD} = {180^ \circ } - {30^ \circ } = {150^ \circ }\end{array}\)Ta có: \(\overrightarrow {BC} .\overrightarrow {CA}  = \overrightarrow {CD} .\overrightarrow {CA}  = CD.CA.\cos \left( {\overrightarrow {CD} ,\overrightarrow {CA} } \right) = 2.2.\cos {150^ \circ } =  - 3\)Chọn D.

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Ph/hs Tham Gia Nhóm Để Cập Nhật Điểm Thi, Điểm Chuẩn Miễn Phí

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