Bài 2: Định Lí Đảo Và Hệ Quả Của Định Lí Ta-Lét
Hướng dẫn giải Bài 10 (Trang 63 SGK Toán Hình học 8, Tập 2)
<p><span id="MathJax-Element-1-Frame" class="mjx-chtml MathJax_CHTML" style="margin: 0px; padding: 1px 0px; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 16.94px; letter-spacing: normal; overflow-wrap: 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"><mo>&#x2206;</mo><mi>A</mi><mi>B</mi><mi>C</mi></math>"><span id="MJXc-Node-1" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2" class="mjx-mrow"><span id="MJXc-Node-3" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">Δ</span></span><span id="MJXc-Node-4" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>C</mi></math></span></span> có đường cao <span id="MathJax-Element-2-Frame" class="mjx-chtml MathJax_CHTML" style="margin: 0px; padding: 1px 0px; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 16.94px; letter-spacing: normal; overflow-wrap: 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>H</mi></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>H</mi></math></span></span>. Đường thẳng <span id="MathJax-Element-3-Frame" class="mjx-chtml MathJax_CHTML" style="margin: 0px; padding: 1px 0px; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 16.94px; letter-spacing: normal; overflow-wrap: 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>d</mi></math>"><span id="MJXc-Node-11" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-12" class="mjx-mrow"><span id="MJXc-Node-13" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">d</span></span></span></span></span> song song với <span id="MathJax-Element-4-Frame" class="mjx-chtml MathJax_CHTML" style="margin: 0px; padding: 1px 0px; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 16.94px; letter-spacing: normal; overflow-wrap: 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>B</mi><mi>C</mi></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>C</mi></math></span></span>, cắt các cạnh <span id="MathJax-Element-5-Frame" class="mjx-chtml MathJax_CHTML" style="margin: 0px; padding: 1px 0px; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 16.94px; letter-spacing: normal; overflow-wrap: 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></math>"><span id="MJXc-Node-18" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-19" class="mjx-mrow"><span id="MJXc-Node-20" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-21" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span><span id="MJXc-Node-22" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-23" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-24" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span></span></span></span> và đường cao <span id="MathJax-Element-6-Frame" class="mjx-chtml MathJax_CHTML" style="margin: 0px; padding: 1px 0px; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 16.94px; letter-spacing: normal; overflow-wrap: 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>H</mi></math>"><span id="MJXc-Node-25" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-26" class="mjx-mrow"><span id="MJXc-Node-27" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math></span></span>theo thứ tự tại các điểm <span id="MathJax-Element-7-Frame" class="mjx-chtml MathJax_CHTML" style="margin: 0px; padding: 1px 0px; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 16.94px; letter-spacing: normal; overflow-wrap: 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"><msup><mi>B</mi><mo>&#x2032;</mo></msup><mo>,</mo><msup><mi>C</mi><mo>&#x2032;</mo></msup></math>"><span id="MJXc-Node-29" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-30" class="mjx-mrow"><span id="MJXc-Node-31" class="mjx-msup"><span class="mjx-base"><span id="MJXc-Node-32" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span></span><span class="mjx-sup"><span id="MJXc-Node-33" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">′</span></span></span></span><span id="MJXc-Node-34" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-35" class="mjx-msup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-36" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span></span><span class="mjx-sup"><span id="MJXc-Node-37" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">′</span></span></span></span></span></span></span> và <span id="MathJax-Element-8-Frame" class="mjx-chtml MathJax_CHTML" style="margin: 0px; padding: 1px 0px; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 16.94px; letter-spacing: normal; overflow-wrap: 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"><msup><mi>H</mi><mo>&#x2032;</mo></msup></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>H</mi><mo>′</mo></msup></math></span></span>(h.16)</p>
<p><img src="https://img.loigiaihay.com/picture/2018/0718/b10-trang-63-sgk-toan-8-t2-c2.jpg" /></p>
<p><strong>LG a.</strong></p>
<p>Chứng minh rằng:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>A</mi><mo>⁢</mo><msup><mi>H</mi><mo>'</mo></msup></mrow><mrow><mi>A</mi><mo>⁢</mo><mi>H</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup></mrow><mrow><mi>B</mi><mo>⁢</mo><mi>C</mi></mrow></mfrac><mo>⁢</mo><mo>.</mo></math><br />Phương pháp giải:<br />Áp dụng: Hệ quả của định lý TaLet.<br />Lời giải chi tiết:<br />vì <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><msup><mi>B</mi><mo>'</mo></msup><msup><mi>C</mi><mo>'</mo></msup><mo>/</mo><mo>/</mo><mi>B</mi><mi>C</mi><mo>⇒</mo><mfrac><mrow><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup></mrow><mrow><mi>B</mi><mo>⁢</mo><mi>C</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>A</mi><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup></mrow><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi></mrow></mfrac></mstyle></math> (1) (theo hệ quả định lý TaLet)<br />Trong <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>A</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>H</mi></mstyle></math> có <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><msup><mi>B</mi><mo>'</mo></msup><msup><mi>H</mi><mo>'</mo></msup><mo>/</mo><mo>/</mo><mi>B</mi><mi>H</mi><mo>⇒</mo><mfrac><mrow><mi>A</mi><mo>⁢</mo><msup><mi>H</mi><mo>'</mo></msup></mrow><mrow><mi>A</mi><mo>⁢</mo><mi>H</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>A</mi><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup></mrow><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi></mrow></mfrac></mstyle></math> (2) (theo hệ quả định lý TaLet)<br />Từ (1) và (2)<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup></mrow><mrow><mi>B</mi><mo>⁢</mo><mi>C</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>A</mi><mo>⁢</mo><msup><mi>H</mi><mo>'</mo></msup></mrow><mrow><mi>A</mi><mo>⁢</mo><mi>H</mi></mrow></mfrac><mo>⁢</mo></math></p>
<p><strong>LG b.</strong><br />Áp dụng: Cho biết <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi>A</mi><mo>⁢</mo><msup><mi>H</mi><mo>'</mo></msup><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>⁢</mo><mi>A</mi><mo>⁢</mo><mi>H</mi></mstyle></math> và diện tích <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi mathvariant="normal">△</mi><mo>⁢</mo><mi>A</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>C</mi></mstyle></math> là <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mn>67</mn><mo>,</mo><mpadded><mn>5</mn></mpadded><mo>⁢</mo><msup><mi>cm</mi><mn>2</mn></msup></mstyle></math> Tính diện tích <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>A</mi><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup><mo>.</mo></math><br /><strong>Phương pháp giải:</strong><br />Áp dụng: Hệ quả của định lý TaLet và công thức tính diện tích tam giác.<br /><strong>Lời giải chi tiết:</strong><br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><msup><mi>B</mi><mo>'</mo></msup><msup><mi>C</mi><mo>'</mo></msup><mo>/</mo><mo>/</mo><mi>B</mi><mi>C</mi></mstyle></math> mà <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi>A</mi><mo>⁢</mo><mi>H</mi><mo>⟂</mo><mi>B</mi><mo>⁢</mo><mi>C</mi></mstyle></math> nên <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi>A</mi><mo>⁢</mo><msup><mi>H</mi><mo>'</mo></msup><mo>⟂</mo><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup></mstyle></math> hay <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi>A</mi><mo>⁢</mo><msup><mi>H</mi><mo>'</mo></msup></mstyle></math> là đường cao của <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>A</mi><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup></mstyle></math><br />Giả thiết: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>⁢</mo><msup><mi>H</mi><mo>'</mo></msup><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>⁢</mo><mi>A</mi><mo>⁢</mo><mi>H</mi><mo>.</mo></math></p>
<p>Áp dụng kết quả câu a) ta có:<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup></mrow><mrow><mi>B</mi><mo>⁢</mo><mi>C</mi></mrow></mfrac><mo>=</mo><mo>⁢</mo><mfrac><mrow><mi>A</mi><mo>⁢</mo><msup><mi>H</mi><mo>'</mo></msup></mrow><mrow><mi>A</mi><mo>⁢</mo><mi>H</mi></mrow></mfrac><mo>=</mo><mpadded><mfrac><mn>1</mn><mn>3</mn></mfrac></mpadded><mspace linebreak="newline"/><mspace linebreak="newline"/><mrow><mo>⇒</mo><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup><mo>⁢</mo><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mpadded><mi>C</mi></mpadded></mrow><mspace linebreak="newline"/><msub><mi>S</mi><mrow><mi>A</mi><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup></mrow></msub><mo>⁢</mo><mo>=</mo><mfrac><mstyle displaystyle="true"><mn>1</mn></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle></mfrac><mo>⁢</mo><mi>A</mi><mo>⁢</mo><msup><mi>H</mi><mo>'</mo></msup><mo>⋅</mo><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><mpadded><msup><mi>C</mi><mo>'</mo></msup></mpadded><mspace linebreak="newline"/><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><mfrac><mstyle displaystyle="true"><mn>1</mn></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle></mfrac><mo>⋅</mo><mfrac><mstyle displaystyle="true"><mn>1</mn></mstyle><mstyle displaystyle="true"><mn>3</mn></mstyle></mfrac><mo>⁢</mo><mi>A</mi><mo>⁢</mo><mi>H</mi><mo>⋅</mo><mfrac><mstyle displaystyle="true"><mn>1</mn></mstyle><mstyle displaystyle="true"><mn>3</mn></mstyle></mfrac><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mpadded><mi>C</mi></mpadded><mspace linebreak="newline"/><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><mfrac><mstyle displaystyle="true"><mn>1</mn></mstyle><mstyle displaystyle="true"><mn>9</mn></mstyle></mfrac><mo>⋅</mo><mrow><mo>(</mo><mfrac><mstyle displaystyle="true"><mn>1</mn></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle></mfrac><mo>⁢</mo><mi>A</mi><mo>⁢</mo><mi>H</mi><mo>⋅</mo><mi>B</mi><mo>⁢</mo><mi>C</mi><mo>)</mo></mrow><mspace linebreak="newline"/><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><mfrac><mstyle displaystyle="true"><mn>1</mn></mstyle><mstyle displaystyle="true"><mn>9</mn></mstyle></mfrac><mo>⋅</mo><mpadded><msub><mi>S</mi><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>C</mi></mrow></msub></mpadded><mspace linebreak="newline"/><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><mfrac><mstyle displaystyle="true"><mn>1</mn></mstyle><mstyle displaystyle="true"><mn>9</mn></mstyle></mfrac><mo>⋅</mo><mn>67</mn><mo>,</mo><mn>5</mn><mo>=</mo><mn>7</mn><mo>,</mo><mpadded><mn>5</mn></mpadded><mo>⁢</mo><msup><mi>cm</mi><mn>2</mn></msup><mspace linebreak="newline"/></math></p>
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