Bài 6. Trường hợp đồng dạng thứ hai
Hướng dẫn giải Bài 33 (Trang 77 SGK Toán Hình học 8, Tập 2)
<p><strong class="content_question">Đề bài</strong></p>
<p>Chứng minh rằng nếu tam giác <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"><msup><mi>A</mi><mo>&#x2032;</mo></msup><msup><mi>B</mi><mo>&#x2032;</mo></msup><msup><mi>C</mi><mo>&#x2032;</mo></msup></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-msup"><span class="mjx-base"><span id="MJXc-Node-4" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span></span><span class="mjx-sup"><span id="MJXc-Node-5" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">′</span></span></span></span><span id="MJXc-Node-6" class="mjx-msup"><span class="mjx-base"><span id="MJXc-Node-7" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span></span><span class="mjx-sup"><span id="MJXc-Node-8" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">′</span></span></span></span><span id="MJXc-Node-9" class="mjx-msup"><span class="mjx-base"><span id="MJXc-Node-10" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span></span><span class="mjx-sup"><span id="MJXc-Node-11" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">′</span></span></span></span></span></span></span> đồng dạng với tam giác <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>B</mi><mi>C</mi></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mi>C</mi></math></span></span> theo tỉ số <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>k</mi></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span>, thì tỉ số của hai đường trung tuyến tương ứng với hai tam giác đó cũng bằng <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>k</mi></math>"><span id="MJXc-Node-20" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-21" class="mjx-mrow"><span id="MJXc-Node-22" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">k</span></span></span></span></span>.</p>
<p><strong class="content_detail">Lời giải chi tiết</strong></p>
<p><strong class="content_detail"><img src="https://img.loigiaihay.com/picture/2018/0718/b33-trang-77-sgk-toan-8-t2-c2.jpg" /></strong></p>
<p><span class="content_detail">Giả sử <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup></mstyle></math> đồng dạng <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> theo tỉ số <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi>k</mi><mo>,</mo><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>M</mi><mo>'</mo></msup><mo>,</mo><mi>A</mi><mo>⁢</mo><mi>M</mi></mstyle></math> là hai đường trung tuyến tương ứng.<br />vì <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup></mstyle></math> đồng dạng <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> theo tỉ số k (giả thiết)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mfrac><mrow><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup></mrow><mrow><mi>A</mi><mo>⁢</mo><mi>B</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><mi>k</mi></mstyle></math> (tính chất hai tam giác đồng dạng)<br />Mà <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup><mo>=</mo><mn>2</mn><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>M</mi><mo>'</mo></msup><mo>,</mo><mi>B</mi><mo>⁢</mo><mi>C</mi><mo>=</mo><mn>2</mn><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>M</mi></mstyle></math> (tính chất trung tuyến)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mfrac><mrow><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup></mrow><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>M</mi><mo>'</mo></msup></mrow><mrow><mn>2</mn><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>M</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>M</mi><mo>'</mo></msup></mrow><mrow><mi>B</mi><mo>⁢</mo><mi>M</mi></mrow></mfrac></mstyle></math><br />Xét <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>M</mi></mstyle></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi mathvariant="normal">△</mi><mo>⁢</mo><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>M</mi><mo>'</mo></msup></mstyle></math> có:<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mi>B</mi><mo>^</mo></mover><mo>=</mo><mover accent="true"><msup><mi>B</mi><mo>'</mo></msup><mo>^</mo></mover></mstyle></math> (vì <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup></mstyle></math> đồng dạng <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>C</mi></mstyle></math>)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mfrac><mrow><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup></mrow><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>M</mi><mo>'</mo></msup></mrow><mrow><mi>B</mi><mo>⁢</mo><mi>M</mi></mrow></mfrac></mstyle></math> (chứng minh trên)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>M</mi><mo>'</mo></msup></mstyle></math> đồng dạng <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>M</mi></mstyle></math> theo tỉ số <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mfrac><mrow><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup></mrow><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi></mrow></mfrac><mo>=</mo><mi>k</mi></mstyle></math> (c-g-c)<br /></span><span class="content_detail"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mfrac><mrow><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>M</mi><mo>'</mo></msup></mrow><mrow><mi>A</mi><mo>⁢</mo><mi>M</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup></mrow><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi></mrow></mfrac><mo>=</mo><mi>k</mi></mstyle></math>. (tính chất hai tam giác đồng dạng)</span><strong class="content_detail"><br /></strong></p>
<p> </p>
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