Hướng dẫn giải Bài 4 (Trang 92 SGK Toán Hình học 11)

Cho hình tứ diện ABCD. Gọi M và N lần lượt là trung điểm của AB và CD. Chứng minh rằng: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">a</mi><mo>)</mo><mo> </mo><mover><mrow><mi>M</mi><mi>N</mi></mrow><mo>→</mo></mover><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mfenced><mrow><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>B</mi><mi>C</mi></mrow><mo>→</mo></mover></mrow></mfenced><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi mathvariant="normal">b</mi><mo>)</mo><mo> </mo><mover><mrow><mi>M</mi><mi>N</mi></mrow><mo>→</mo></mover><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mfenced><mrow><mover><mrow><mi>A</mi><mi>C</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover></mrow></mfenced><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo></math> Giải: <img class="wscnph" src="https://static.colearn.vn:8413/v1.0/upload/library/24022022/f0425b67-ad6a-4dba-b49f-de034ae0c72b.PNG" /> a) Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mover><mrow><mi>M</mi><mi>N</mi></mrow><mo>→</mo></mover><mo>=</mo><mover><mrow><mi>M</mi><mi>A</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>D</mi><mi>N</mi></mrow><mo>→</mo></mover></mtd></mtr><mtr><mtd><mover><mrow><mi>M</mi><mi>N</mi></mrow><mo>→</mo></mover><mo>=</mo><mover><mrow><mi>M</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>B</mi><mi>C</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>C</mi><mi>N</mi></mrow><mo>→</mo></mover></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mi>Suy</mi><mo> </mo><mi>ra</mi><mo> </mo><mn>2</mn><mover><mrow><mi>M</mi><mi>N</mi></mrow><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mover><mrow><mi>M</mi><mi>A</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>M</mi><mi>B</mi></mrow><mo>→</mo></mover></mrow></mfenced><mo>+</mo><mfenced><mrow><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>B</mi><mi>C</mi></mrow><mo>→</mo></mover></mrow></mfenced><mo>+</mo><mfenced><mrow><mover><mrow><mi>D</mi><mi>N</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>C</mi><mi>N</mi></mrow><mo>→</mo></mover></mrow></mfenced><mo>=</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>B</mi><mi>C</mi></mrow><mo>→</mo></mover><mspace linebreak="newline"/><mo>⇒</mo><mover><mrow><mi>M</mi><mi>N</mi></mrow><mo>→</mo></mover><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mfenced><mrow><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>B</mi><mi>C</mi></mrow><mo>→</mo></mover></mrow></mfenced></math> b) Tương tự: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mover><mrow><mi>M</mi><mi>N</mi></mrow><mo>→</mo></mover><mo>=</mo><mover><mrow><mi>M</mi><mi>A</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>C</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>C</mi><mi>N</mi></mrow><mo>→</mo></mover></mtd></mtr><mtr><mtd><mover><mrow><mi>M</mi><mi>N</mi></mrow><mo>→</mo></mover><mo>=</mo><mover><mrow><mi>M</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>D</mi><mi>N</mi></mrow><mo>→</mo></mover></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mi>Suy</mi><mo> </mo><mi>ra</mi><mo> </mo><mn>2</mn><mover><mrow><mi>M</mi><mi>N</mi></mrow><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mover><mrow><mi>M</mi><mi>A</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>M</mi><mi>B</mi></mrow><mo>→</mo></mover></mrow></mfenced><mo>+</mo><mfenced><mrow><mover><mrow><mi>A</mi><mi>C</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover></mrow></mfenced><mo>+</mo><mfenced><mrow><mover><mrow><mi>C</mi><mi>N</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>D</mi><mi>N</mi></mrow><mo>→</mo></mover></mrow></mfenced><mo>=</mo><mover><mrow><mi>A</mi><mi>C</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover><mspace linebreak="newline"/><mo>⇒</mo><mover><mrow><mi>M</mi><mi>N</mi></mrow><mo>→</mo></mover><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mfenced><mrow><mover><mrow><mi>A</mi><mi>C</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover></mrow></mfenced></math>

Giải bài tập 4 ( SGK Toán 11, trang 92, Hình học)

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