Hướng dẫn giải Bài 5 (Trang 99 SGK Toán Hình học 12)

Cho tứ diện ABCD có cạnh AD vuông góc với mặt phẳng (ABC). Biết rằng AC = AD = 4cm, AB = 3cm, BC = 5 cm. a) Tính thể tích tứ diện ABCD. b) Tính khoảng cách từ điểm A tới mặt phẳng (BCD). Giải: <img class="wscnph" src="https://static.colearn.vn:8413/v1.0/upload/library/18022022/372a0fb5-1860-48cb-9671-fe239b30cc1f.PNG" /> a) Chọn hệ toạ độ gốc là điểm <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>, các đường thẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mo>,</mo><mi>A</mi><mi>C</mi><mo>,</mo><mi>A</mi><mi>D</mi></math> theo thứ tự là các trục <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>x</mi><mo>,</mo><mo> </mo><mi>O</mi><mi>y</mi><mo>,</mo><mo> </mo><mi>O</mi><mi>z</mi><mo>.</mo></math><span id="MathJax-Element-2-Frame" class="mjx-chtml MathJax_CHTML" style="box-sizing: border-box; margin: 0px; padding: 1px 0px; overflow-wrap: normal; word-break: break-word; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 18.08px; letter-spacing: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mo>,</mo><mi>A</mi><mi>C</mi><mo>,</mo><mi>A</mi><mi>D</mi></math>"><span id="MathJax-Element-4-Frame" class="mjx-chtml MathJax_CHTML" style="box-sizing: border-box; margin: 0px; padding: 1px 0px; overflow-wrap: normal; word-break: break-word; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 18.08px; letter-spacing: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo stretchy="false">(</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>B</mi><mo stretchy="false">(</mo><mn>3</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo stretchy="false">)</mo><mo>;</mo><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo>;</mo><mn>4</mn><mo>;</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>D</mi><mo stretchy="false">(</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>4</mn><mo stretchy="false">)</mo></math>"> <span class="mjx-chtml MathJax_CHTML" style="box-sizing: border-box; margin: 0px; padding: 1px 0px; overflow-wrap: normal; word-break: break-word; display: inline-table; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: 400; font-size: 18.08px; letter-spacing: normal; word-spacing: 0px; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; color: #131313; font-family: Quicksand; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; background-color: #ffffff; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo stretchy="false">(</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>B</mi><mo stretchy="false">(</mo><mn>3</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo stretchy="false">)</mo><mo>;</mo><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo>;</mo><mn>4</mn><mo>;</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>D</mi><mo stretchy="false">(</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>4</mn><mo stretchy="false">)</mo></math>">Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mi>A</mi><mo>(</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>)</mo><mo>,</mo><mi>B</mi><mo>(</mo><mn>3</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>)</mo><mo>;</mo><mi>C</mi><mo>(</mo><mn>0</mn><mo>;</mo><mn>4</mn><mo>;</mo><mn>0</mn><mo>)</mo><mo>,</mo><mi>D</mi><mo>(</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>4</mn><mo>)</mo></math> Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>=</mo><mo>(</mo><mn>3</mn><mo>;</mo><mo> </mo><mn>0</mn><mo>;</mo><mo> </mo><mn>0</mn><mo>)</mo><mo> </mo><mo>⇒</mo><mi>A</mi><mi>B</mi><mo> </mo><mo>=</mo><mo> </mo><mn>3</mn><mo>;</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>C</mi></mrow><mo>→</mo></mover><mo> </mo><mo>=</mo><mo> </mo><mo>(</mo><mn>0</mn><mo>;</mo><mo> </mo><mn>4</mn><mo>;</mo><mo> </mo><mn>0</mn><mo>)</mo><mo> </mo><mo>⇒</mo><mi>A</mi><mi>C</mi><mo> </mo><mo>=</mo><mo> </mo><mn>4</mn><mo>;</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo> </mo><mo>=</mo><mo> </mo><mo>(</mo><mn>0</mn><mo>;</mo><mo> </mo><mn>0</mn><mo>;</mo><mo> </mo><mn>4</mn><mo>)</mo><mo> </mo><mo>⇒</mo><mi>A</mi><mi>D</mi><mo> </mo><mo>=</mo><mo> </mo><mn>4</mn><mo>;</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mrow><mi>A</mi><mi>B</mi><mi>C</mi><mi>D</mi></mrow></msub><mo>=</mo><mo> </mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mi>A</mi><mi>B</mi><mo>.</mo><mi>A</mi><mi>C</mi><mo>.</mo><mi>A</mi><mi>D</mi><mo>=</mo><mn>8</mn><mo>(</mo><mi>c</mi><msup><mi>m</mi><mn>3</mn></msup><mo>)</mo></math> <span class="mjx-chtml MathJax_CHTML" style="box-sizing: border-box; margin: 0px; padding: 1px 0px; overflow-wrap: normal; word-break: break-word; display: inline-table; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-size: 18.08px; letter-spacing: normal; word-spacing: 0px; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; color: #131313; font-family: Quicksand; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; background-color: #ffffff; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo stretchy="false">(</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>B</mi><mo stretchy="false">(</mo><mn>3</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo stretchy="false">)</mo><mo>;</mo><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo>;</mo><mn>4</mn><mo>;</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>D</mi><mo stretchy="false">(</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>4</mn><mo stretchy="false">)</mo></math>">b)Áp dụng công thức phương trình mặt phẳng theo đoạn chắn, ta có phương trình mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>B</mi><mi>D</mi><mi>C</mi><mo>)</mo></math> là:  <span class="mjx-chtml MathJax_CHTML" style="box-sizing: border-box; margin: 0px; padding: 1px 0px; overflow-wrap: normal; word-break: break-word; display: inline-table; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-size: 18.08px; letter-spacing: normal; word-spacing: 0px; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; color: #131313; font-family: Quicksand; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; background-color: #ffffff; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo stretchy="false">(</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>B</mi><mo stretchy="false">(</mo><mn>3</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo stretchy="false">)</mo><mo>;</mo><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo>;</mo><mn>4</mn><mo>;</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>D</mi><mo stretchy="false">(</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>4</mn><mo stretchy="false">)</mo></math>"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mn>3</mn></mfrac><mo>+</mo><mfrac><mi>y</mi><mn>4</mn></mfrac><mo>+</mo><mfrac><mi>z</mi><mn>4</mn></mfrac><mo>=</mo><mn>1</mn><mo>⇔</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>z</mi><mo>−</mo><mn>12</mn><mo>=</mo><mn>0</mn></math> <span class="mjx-chtml MathJax_CHTML" style="box-sizing: border-box; margin: 0px; padding: 1px 0px; overflow-wrap: normal; word-break: break-word; display: inline-table; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-size: 18.08px; letter-spacing: normal; word-spacing: 0px; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; color: #131313; font-family: Quicksand; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; background-color: #ffffff; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo stretchy="false">(</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>B</mi><mo stretchy="false">(</mo><mn>3</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo stretchy="false">)</mo><mo>;</mo><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo>;</mo><mn>4</mn><mo>;</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>D</mi><mo stretchy="false">(</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>4</mn><mo stretchy="false">)</mo></math>">Từ đây ta có: <span class="mjx-chtml MathJax_CHTML" style="box-sizing: border-box; margin: 0px; padding: 1px 0px; overflow-wrap: normal; word-break: break-word; display: inline-table; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-size: 18.08px; letter-spacing: normal; word-spacing: 0px; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; color: #131313; font-family: Quicksand; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; background-color: #ffffff; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo stretchy="false">(</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>B</mi><mo stretchy="false">(</mo><mn>3</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo stretchy="false">)</mo><mo>;</mo><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo>;</mo><mn>4</mn><mo>;</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>D</mi><mo stretchy="false">(</mo><mn>0</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>4</mn><mo stretchy="false">)</mo></math>"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>(</mo><mi>A</mi><mo>,</mo><mo>(</mo><mi>B</mi><mi>D</mi><mi>C</mi><mo>)</mo><mo>)</mo><mo>=</mo><mfrac><mfenced open="|" close="|"><mn>12</mn></mfenced><msqrt><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><msup><mn>3</mn><mn>2</mn></msup><mo>+</mo><msup><mn>3</mn><mn>2</mn></msup></msqrt></mfrac><mo> </mo><mo>=</mo><mfrac><mn>12</mn><msqrt><mn>34</mn></msqrt></mfrac></math></mo></math>

Hướng dẫn Giải Bài 5 (trang 99, SGK Toán 12, Hình học)

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