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Giải bài 4.66 trang 71 sách bài tập toán 10 - Kết nối tri thức với cuộc sống

Cho bốn điểm A, B, C, D trong mặt phẳng. Chứng minh rằng

Tổng hợp đề th🗹i họcဣ 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 bốn điểm \(A,\,\,B,\,\,C,\,\,D\) trong mặt phẳng. Chứng minh rằng \(\overrightarrow {AB} .\overrightarrow {CD}  + \overrightarrow {BC} .\overrightarrow {AD}  + \overrightarrow {CA} .\overrightarrow {BD}  = 0.\)

Lời giải chi tiết

Ta có: \(\overrightarrow {AD}  = \overrightarrow {AB}  + \overrightarrow {BC}  + \overrightarrow {CD} \)\(\overrightarrow {CA}  = \overrightarrow {BA}  - \overrightarrow {BC} \)\(\overrightarrow {BD}  = \overrightarrow {BC}  + \overrightarrow {CD} \)Ta có:\(\overrightarrow {AB} .\overrightarrow {CD}  + \overrightarrow {BC} .\overrightarrow {AD}  + \overrightarrow {CA} .\overrightarrow {BD}  = \overrightarrow {AB} .\overrightarrow {CD}  + \overrightarrow {BC} \left( {\overrightarrow {AB}  + \overrightarrow {BC}  + \overrightarrow {CD} } \right) + \left( {\overrightarrow {BA}  - \overrightarrow {BC} } \right)\left( {\overrightarrow {BC}  + \overrightarrow {CD} } \right)\)\( = \overrightarrow {AB} .\overrightarrow {CD}  + \overrightarrow {BC} .\overrightarrow {AB}  + {\overrightarrow {BC} ^2} + \overrightarrow {BC} .\overrightarrow {CD}  + \overrightarrow {BA} .\overrightarrow {BC}  + \overrightarrow {BA} .\overrightarrow {CD}  - {\overrightarrow {BC} ^2} - \overrightarrow {BC} .\overrightarrow {CD} \)
\( = \left( {\overrightarrow {AB} .\overrightarrow {CD}  + \overrightarrow {BA} .\overrightarrow {CD} } \right) + \left( {\overrightarrow {BC} .\overrightarrow {AB}  + \overrightarrow {BA} .\overrightarrow {BC} } \right) + \left( {{{\overrightarrow {BC} }^2} - {{\overrightarrow {BC} }^2}} \right) + \left( {\overrightarrow {BC} .\overrightarrow {CD}  - \overrightarrow {BC} .\overrightarrow {CD} } \right) = 0\)\( \Rightarrow \,\,\overrightarrow {AB} .\overrightarrow {CD}  + \overrightarrow {BC} .\overrightarrow {AD}  + \overrightarrow {CA} .\overrightarrow {BD}  = 0\) (đpcm)

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