Bài tập cuối chương III
Hướng dẫn giải Bài 3.16 (Trang 44 SGK Toán 10, Bộ Kết nối tri thức, Tập 1)
<p><em><strong>Cho tam giác ABC có trung tuyến AM. Chứng minh rằng:</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi><mo mathvariant="bold">)</mo><mo mathvariant="bold"> </mo><mi mathvariant="bold-italic">c</mi><mi mathvariant="bold-italic">o</mi><mi mathvariant="bold-italic">s</mi><mover><mrow><mi mathvariant="bold">A</mi><mi mathvariant="bold">M</mi><mi mathvariant="bold">B</mi></mrow><mo mathvariant="bold">^</mo></mover><mo mathvariant="bold">+</mo><mi mathvariant="bold-italic">c</mi><mi mathvariant="bold-italic">o</mi><mi mathvariant="bold-italic">s</mi><mover><mrow><mi mathvariant="bold">A</mi><mi mathvariant="bold">M</mi><mi mathvariant="bold">C</mi></mrow><mo mathvariant="bold">^</mo></mover><mo mathvariant="bold">=</mo><mn mathvariant="bold">0</mn><mo mathvariant="bold">;</mo><mspace linebreak="newline"/><mi mathvariant="bold-italic">b</mi><mo mathvariant="bold">)</mo><mo mathvariant="bold"> </mo><mi mathvariant="bold-italic">M</mi><msup><mi mathvariant="bold-italic">A</mi><mn mathvariant="bold">2</mn></msup><mo mathvariant="bold">+</mo><mi mathvariant="bold-italic">M</mi><msup><mi mathvariant="bold-italic">B</mi><mn mathvariant="bold">2</mn></msup><mo mathvariant="bold">-</mo><mi mathvariant="bold-italic">A</mi><msup><mi mathvariant="bold-italic">B</mi><mn mathvariant="bold">2</mn></msup><mo mathvariant="bold">=</mo><mn mathvariant="bold">2</mn><mi mathvariant="bold-italic">M</mi><mi mathvariant="bold-italic">A</mi><mo mathvariant="bold">.</mo><mi mathvariant="bold-italic">M</mi><mi mathvariant="bold-italic">B</mi><mo mathvariant="bold">.</mo><mi mathvariant="bold-italic">c</mi><mi mathvariant="bold-italic">o</mi><mi mathvariant="bold-italic">s</mi><mover><mrow><mi mathvariant="bold">A</mi><mi mathvariant="bold">M</mi><mi mathvariant="bold">B</mi></mrow><mo mathvariant="bold">^</mo></mover><mo mathvariant="bold"> </mo><mspace linebreak="newline"/><mi mathvariant="bold-italic">v</mi><mi mathvariant="bold-italic">à</mi><mo mathvariant="bold"> </mo><mi mathvariant="bold-italic">M</mi><msup><mi mathvariant="bold-italic">A</mi><mn mathvariant="bold">2</mn></msup><mo mathvariant="bold">+</mo><mi mathvariant="bold-italic">M</mi><msup><mi mathvariant="bold-italic">C</mi><mn mathvariant="bold">2</mn></msup><mo mathvariant="bold">-</mo><mi mathvariant="bold-italic">A</mi><msup><mi mathvariant="bold-italic">C</mi><mn mathvariant="bold">2</mn></msup><mo mathvariant="bold">=</mo><mn mathvariant="bold">2</mn><mi mathvariant="bold-italic">M</mi><mi mathvariant="bold-italic">A</mi><mo mathvariant="bold">.</mo><mi mathvariant="bold-italic">M</mi><mi mathvariant="bold-italic">C</mi><mo mathvariant="bold">.</mo><mi mathvariant="bold-italic">c</mi><mi mathvariant="bold-italic">o</mi><mi mathvariant="bold-italic">s</mi><mover><mrow><mi mathvariant="bold">A</mi><mi mathvariant="bold">M</mi><mi mathvariant="bold">C</mi></mrow><mo mathvariant="bold">^</mo></mover><mspace linebreak="newline"/><mi mathvariant="bold-italic">c</mi><mo mathvariant="bold">)</mo><mo mathvariant="bold"> </mo><mi mathvariant="bold-italic">M</mi><msup><mi mathvariant="bold-italic">A</mi><mn mathvariant="bold">2</mn></msup><mo mathvariant="bold">=</mo><mfrac><mrow><mn mathvariant="bold">2</mn><mo mathvariant="bold">(</mo><mi mathvariant="bold">A</mi><msup><mi mathvariant="bold">B</mi><mn mathvariant="bold">2</mn></msup><mo mathvariant="bold">+</mo><mi mathvariant="bold">A</mi><msup><mi mathvariant="bold">C</mi><mn mathvariant="bold">2</mn></msup><mo mathvariant="bold">)</mo><mo mathvariant="bold">-</mo><mi mathvariant="bold">B</mi><msup><mi mathvariant="bold">C</mi><mn mathvariant="bold">2</mn></msup></mrow><mn mathvariant="bold">4</mn></mfrac><mo mathvariant="bold"> </mo><mo mathvariant="bold">(</mo><mi mathvariant="bold-italic">c</mi><mi mathvariant="bold-italic">ô</mi><mi mathvariant="bold-italic">n</mi><mi mathvariant="bold-italic">g</mi><mo mathvariant="bold"> </mo><mi mathvariant="bold-italic">t</mi><mi mathvariant="bold-italic">h</mi><mi mathvariant="bold-italic">ứ</mi><mi mathvariant="bold-italic">c</mi><mo mathvariant="bold"> </mo><mi mathvariant="bold-italic">đ</mi><mi mathvariant="bold-italic">ư</mi><mi mathvariant="bold-italic">ờ</mi><mi mathvariant="bold-italic">n</mi><mi mathvariant="bold-italic">g</mi><mo mathvariant="bold"> </mo><mi mathvariant="bold-italic">t</mi><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">u</mi><mi mathvariant="bold-italic">n</mi><mi mathvariant="bold-italic">g</mi><mo mathvariant="bold"> </mo><mi mathvariant="bold-italic">t</mi><mi mathvariant="bold-italic">u</mi><mi mathvariant="bold-italic">y</mi><mi mathvariant="bold-italic">ế</mi><mi mathvariant="bold-italic">n</mi><mo mathvariant="bold">)</mo><mo mathvariant="bold">.</mo></math></p>
<p><span style="text-decoration: underline;"><em><strong>Lời giải:</strong></em></span></p>
<p><img class="wscnph" style="max-width: 100%;" src="https://static.colearn.vn:8413/v1.0/upload/library/15062022/2-ZompLK.png" width="270" height="220" /></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>B</mi></mrow><mo>^</mo></mover><mo>+</mo><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>C</mi></mrow><mo>^</mo></mover><mo>=</mo><mn>0</mn><mspace linebreak="newline"/><mi>T</mi><mi>a</mi><mo> </mo><mi>c</mi><mi>ó</mi><mo>:</mo><mo> </mo><mover><mrow><mi>A</mi><mi>M</mi><mi>B</mi></mrow><mo>^</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>M</mi><mi>C</mi></mrow><mo>^</mo></mover><mo>=</mo><mn>180</mn><mo>°</mo><mspace linebreak="newline"/><mover><mrow><mi>A</mi><mi>M</mi><mi>C</mi></mrow><mo>^</mo></mover><mo>=</mo><mn>180</mn><mo>°</mo><mo>-</mo><mover><mrow><mi>A</mi><mi>M</mi><mi>B</mi></mrow><mo>^</mo></mover><mspace linebreak="newline"/><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>B</mi></mrow><mo>^</mo></mover><mo>=</mo><mo>-</mo><mi>cos</mi><mo>(</mo><mn>180</mn><mo>°</mo><mo>-</mo><mover><mrow><mi>A</mi><mi>M</mi><mi>B</mi></mrow><mo>^</mo></mover><mo>)</mo><mo>=</mo><mo>-</mo><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>C</mi></mrow><mo>^</mo></mover><mspace linebreak="newline"/><mo>⇒</mo><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>B</mi></mrow><mo>^</mo></mover><mo>+</mo><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>C</mi></mrow><mo>^</mo></mover><mo>=</mo><mo>-</mo><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>C</mi></mrow><mo>^</mo></mover><mo>+</mo><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>C</mi></mrow><mo>^</mo></mover><mo>=</mo><mn>0</mn></math></p>
<p>b) Xét ΔAMB, ta có:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><msup><mi>B</mi><mn>2</mn></msup><mo>=</mo><mi>M</mi><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><mi>M</mi><msup><mi>B</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>M</mi><mi>A</mi><mo>.</mo><mi>M</mi><mi>B</mi><mo>.</mo><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>B</mi></mrow><mo>^</mo></mover><mspace linebreak="newline"/><mo>⇔</mo><mi>M</mi><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><mi>M</mi><msup><mi>B</mi><mn>2</mn></msup><mo>-</mo><mi>A</mi><msup><mi>B</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><mi>M</mi><mi>A</mi><mo>.</mo><mi>M</mi><mi>B</mi><mo>.</mo><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>B</mi></mrow><mo>^</mo></mover><mo> </mo><mo>(</mo><mn>1</mn><mo>)</mo></math></p>
<p>Xét ΔAMC, ta có:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><msup><mi>C</mi><mn>2</mn></msup><mo>=</mo><mi>M</mi><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><mi>M</mi><msup><mi>C</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>M</mi><mi>A</mi><mo>.</mo><mi>M</mi><mi>C</mi><mo>.</mo><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>C</mi></mrow><mo>^</mo></mover><mspace linebreak="newline"/><mo>⇔</mo><mi>M</mi><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><mi>M</mi><msup><mi>C</mi><mn>2</mn></msup><mo>-</mo><mi>A</mi><msup><mi>C</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><mi>M</mi><mi>A</mi><mo>.</mo><mi>M</mi><mi>C</mi><mo>.</mo><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>C</mi></mrow><mo>^</mo></mover><mo> </mo><mo>(</mo><mn>2</mn><mo>)</mo></math></p>
<p>c) Cộng vế với vế của (1) với (2), ta được:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><mi>M</mi><msup><mi>B</mi><mn>2</mn></msup><mo>-</mo><mi>A</mi><msup><mi>B</mi><mn>2</mn></msup><mo>+</mo><mi>M</mi><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><mi>M</mi><msup><mi>C</mi><mn>2</mn></msup><mo>-</mo><mi>A</mi><msup><mi>C</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><mi>M</mi><mi>A</mi><mo>.</mo><mi>M</mi><mi>B</mi><mo>.</mo><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>B</mi></mrow><mo>^</mo></mover><mo>+</mo><mn>2</mn><mi>M</mi><mi>A</mi><mo>.</mo><mi>M</mi><mi>B</mi><mo>.</mo><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>C</mi></mrow><mo>^</mo></mover><mspace linebreak="newline"/><mo>⇔</mo><mn>2</mn><mi>M</mi><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mi>B</mi><msup><mi>C</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>-</mo><mi>A</mi><msup><mi>B</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mi>B</mi><msup><mi>C</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>-</mo><mi>A</mi><msup><mi>C</mi><mn>2</mn></msup><mspace linebreak="newline"/><mo>=</mo><mn>2</mn><mi>M</mi><mi>A</mi><mo>.</mo><mfrac><mrow><mi>B</mi><mi>C</mi></mrow><mn>2</mn></mfrac><mo>.</mo><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>B</mi></mrow><mo>^</mo></mover><mo>+</mo><mn>2</mn><mi>M</mi><mi>A</mi><mo>.</mo><mfrac><mrow><mi>B</mi><mi>C</mi></mrow><mn>2</mn></mfrac><mo>.</mo><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>C</mi></mrow><mo>^</mo></mover><mspace linebreak="newline"/><mo>(</mo><mi>V</mi><mi>ì</mi><mo> </mo><mi>M</mi><mi>B</mi><mo>=</mo><mi>M</mi><mi>C</mi><mo>=</mo><mfrac><mrow><mi>B</mi><mi>C</mi></mrow><mn>2</mn></mfrac><mo>)</mo><mspace linebreak="newline"/><mo>⇔</mo><mn>2</mn><mi>M</mi><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mi>B</mi><msup><mi>C</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo>-</mo><mi>A</mi><msup><mi>B</mi><mn>2</mn></msup><mo>-</mo><mi>A</mi><msup><mi>C</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><mi>M</mi><mi>A</mi><mo>.</mo><mfrac><mrow><mi>B</mi><mi>C</mi></mrow><mn>2</mn></mfrac><mo>.</mo><mo>(</mo><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>B</mi></mrow><mo>^</mo></mover><mo>+</mo><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>C</mi></mrow><mo>^</mo></mover><mo>)</mo><mspace linebreak="newline"/><mo>⇔</mo><mn>2</mn><mi>M</mi><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mi>B</mi><msup><mi>C</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo>-</mo><mi>A</mi><msup><mi>B</mi><mn>2</mn></msup><mo>-</mo><mi>A</mi><msup><mi>C</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn><mo> </mo><mo>(</mo><mi>v</mi><mi>ì</mi><mo> </mo><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>B</mi></mrow><mo>^</mo></mover><mo>+</mo><mi>cos</mi><mover><mrow><mi>A</mi><mi>M</mi><mi>C</mi></mrow><mo>^</mo></mover><mo>=</mo><mn>0</mn><mo>)</mo><mspace linebreak="newline"/><mo>⇔</mo><mn>2</mn><mi>M</mi><msup><mi>A</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mn>2</mn><mi>A</mi><msup><mi>B</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>A</mi><msup><mi>C</mi><mn>2</mn></msup><mo>-</mo><mi>B</mi><msup><mi>C</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mspace linebreak="newline"/><mo>⇔</mo><mi>M</mi><msup><mi>A</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mn>2</mn><mo>(</mo><mi>A</mi><msup><mi>B</mi><mn>2</mn></msup><mo>+</mo><mi>A</mi><msup><mi>C</mi><mn>2</mn></msup><mo>)</mo><mo>-</mo><mi>B</mi><msup><mi>C</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo> </mo><mo>(</mo><mi>c</mi><mi>ô</mi><mi>n</mi><mi>g</mi><mo> </mo><mi>t</mi><mi>h</mi><mi>ứ</mi><mi>c</mi><mo> </mo><mi>đ</mi><mi>ư</mi><mi>ờ</mi><mi>n</mi><mi>g</mi><mo> </mo><mi>t</mi><mi>r</mi><mi>u</mi><mi>n</mi><mi>g</mi><mo> </mo><mi>t</mi><mi>u</mi><mi>y</mi><mi>ế</mi><mi>n</mi><mo>)</mo></math></p>
<p> </p>
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