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Giải bài tập 2.36 trang 84 SGK Toán 12 tập 1 - Cùng khám phá

Tam giác ABC có \(A(1;0;1),B(0;2;3),C(2;1;0)\). Độ dài đường trung tuyến AM là A. \(\frac{1}{2}\). B. \(\frac{{\sqrt {11} }}{2}\). C. \(\frac{{\sqrt {12} }}{2}\). D. \(\frac{{\sqrt {10} }}{2}\).
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Đề bài

Tam giác ABC có \(A(1;0;1),B(0;2;3),C(2;1;0)\). Độ dài đường trung tuyến AM là A. \(\frac{1}{2}\). B. \(\frac{{\sqrt {11} }}{2}\). C. \(\frac{{\sqrt {12} }}{2}\). D. \(\frac{{\sqrt {10} }}{2}\).

Phương pháp giải - Xem chi tiết

- Đầu tiên, tính tọa độ trung điểm \(M\) của cạnh BC: \(M\left( {\frac{{{x_B} + {x_C}}}{2},\frac{{{y_B} + {y_C}}}{2},\frac{{{z_B} + {z_C}}}{2}} \right)\)  - Sau đó, tính độ dài đoạn AM bằng công thức: \(AM = \sqrt {{{({x_A} - {x_M})}^2} + {{({y_A} - {y_M})}^2} + {{({z_A} - {z_M})}^2}} \)

Lời giải chi tiết

- Tọa độ trung điểm \(M\) của BC là: \(M\left( {\frac{{0 + 2}}{2},\frac{{2 + 1}}{2},\frac{{3 + 0}}{2}} \right) = M(1;1.5;1.5)\)  - Độ dài AM: \(AM = \sqrt {{{(1 - 1)}^2} + {{(0 - 1.5)}^2} + {{(1 - 1.5)}^2}}  = \sqrt {0 + 2.25 + 0.25}  = \frac{{\sqrt {10} }}{2}\) Chọn D.

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