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Giải bài 81 trang 108 SBT toán 10 - Cánh diều

Cho tử giác ABCD. M là điểm thay đổi trong mặt phẳng thoả mãn \(\left( {\overrightarrow {MA} + \overrightarrow {MB} } \right).\left( {\overrightarrow {MC} + \overrightarrow {MD} } \right) = 0\). Chứng minh rằng điểm M luôn nằm trên một đường tròn cố định.

Tổng hợp đề🐼 thi học kì 2 lớp 10 tất cả cá༒c môn - Cánh diều

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Đề bài

Cho tử giác ABCD. M là điểm thay đổi trong mặt phẳng thoả mãn \(\left( {\overrightarrow {MA}  + \overrightarrow {MB} } \right).\left( {\overrightarrow {MC}  + \overrightarrow {MD} } \right) = 0\). Chứng minh rằng điểm M luôn nằm trên một đường tròn cố định.

Phương pháp giải - Xem chi tiết

Sử dụng tính chất trung điểm của đoạn thẳng để tìm tập hợp các điểm M

Lời giải chi tiết

Theo giả thiết, \(\left( {\overrightarrow {MA}  + \overrightarrow {MB} } \right).\left( {\overrightarrow {MC}  + \overrightarrow {MD} } \right) = 0\)\( \Leftrightarrow \left[ \begin{array}{l}\overrightarrow {MA}  + \overrightarrow {MB}  = 0\\\overrightarrow {MC}  + \overrightarrow {MD}  = 0\end{array} \right.\)

Gọi P, Q lần lượt là trung điểm của AB và CD \( \Rightarrow \left\{ \begin{array}{l}\overrig🦩htarrow {MA}  + \overrightarrow {MB}  = 2\overrightarrow {MP} \\\overrightarrow {MC}  + \ov♛errightarrow {MD}  = 2\overrightarrow {MQ} \end{array} \right.\)

\( \Rightarrow \left( {\overrightarrow {MA}  + \overrightarrow {MB} } \right).\left( {\overrightarrow {MC}  + \overrightarrow {MD} } \right) = 0 \Leftrightarrow 2\overrightarrow {MP} .2\overrightarrow {MQ}  = 0 \Leftrightarrow \overrightarrow {MP} .\overrightarrow {MQ}  = 0\)

+ Nếu M không trùng với P hoặc Q thì \(\overrightarrow {MP} .\overrightarrow {MQ}♎  = 0 \Leftrightarrow MP \bot MQ\)

\( \Rightarrow \) Tập hợp các điểm M là đường tròn đường kính PQ

+ Nếu M trùng với P hoặc Q thì hiển nhiên M thuộc đường tròn đường kính PQ

Vậy M luôn thuộc đường tròn đường kính PQ cố định

 

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