Bài 5. Trường hợp đồng dạng thứ nhất
Hướng dẫn giải Bài 30 (Trang 75 SGK Toán Hình học 8, Tập 2)
<p><strong class="content_question">Đề bài</strong></p>
<p>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"><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-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-4" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span><span id="MJXc-Node-5" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span></span></span></span> có độ dài các cạnh là <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><mo>=</mo><mn>3</mn><mi>c</mi><mi>m</mi><mo>,</mo><mi>A</mi><mi>C</mi><mo>=</mo><mn>5</mn><mi>c</mi><mi>m</mi><mo>,</mo><mi>B</mi><mi>C</mi><mo>=</mo><mn>7</mn><mi>c</mi><mi>m</mi></math>"><span id="MJXc-Node-6" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7" class="mjx-mrow"><span id="MJXc-Node-8" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-9" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span><span id="MJXc-Node-10" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">=</span></span><span id="MJXc-Node-11" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">3</span></span><span id="MJXc-Node-12" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">c</span></span><span id="MJXc-Node-13" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">m</span></span><span id="MJXc-Node-14" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-15" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-16" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span><span id="MJXc-Node-17" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">=</span></span><span id="MJXc-Node-18" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">5</span></span><span id="MJXc-Node-19" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">c</span></span><span id="MJXc-Node-20" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">m</span></span><span id="MJXc-Node-21" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-22" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I">B</span></span><span id="MJXc-Node-23" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span><span id="MJXc-Node-24" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">=</span></span><span id="MJXc-Node-25" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">7</span></span><span id="MJXc-Node-26" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">c</span></span><span id="MJXc-Node-27" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">m</span></span></span></span></span>. Tam giác <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"><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 class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>A</mi><mo>′</mo></msup><msup><mi>B</mi><mo>′</mo></msup><msup><mi>C</mi><mo>′</mo></msup></math></span></span> đồng dạng với tam giác <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>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> và có chu vi bằng <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"><mn>55</mn><mi>c</mi><mi>m</mi></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>55</mn><mi>c</mi><mi>m</mi></math></span></span>. </p>
<p>Hãy tính độ dài các cạnh của <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"><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-49" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-50" class="mjx-mrow"><span id="MJXc-Node-51" class="mjx-msup"><span class="mjx-base"><span id="MJXc-Node-52" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A'</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>B</mi><mo>′</mo></msup><msup><mi>C</mi><mo>′</mo></msup></math></span></span> (làm tròn đến chữ số thập phân thứ hai).</p>
<p><strong class="content_detail">Lời giải chi tiết</strong></p>
<p><img src="https://img.loigiaihay.com/picture/2019/0218/2019-02-18-160053.jpg" /></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>A</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>C</mi></mstyle></math> đồng dạng <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><mo>⁢</mo><mrow><mo>(</mo><mi>g</mi><mo>⁢</mo><mi>t</mi><mo>)</mo></mrow></mstyle></math><br />Áp dụng tính chất của dãy tỉ số bằng nhau ta có:<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi></mrow><mrow><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>A</mi><mo>⁢</mo><mi>C</mi></mrow><mrow><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>B</mi><mo>⁢</mo><mi>C</mi></mrow><mrow><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi><mo>+</mo><mi>A</mi><mo>⁢</mo><mi>C</mi><mo>+</mo><mi>B</mi><mo>⁢</mo><mi>C</mi></mrow><mrow><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup><mo>+</mo><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup><mo>+</mo><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup></mrow></mfrac><mo>=</mo><mfrac><msub><mi>C</mi><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>C</mi></mrow></msub><msub><mi>C</mi><mrow><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></mrow></msub></mfrac></math><br />hay <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mfrac><mn>3</mn><mrow><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup></mrow></mfrac><mo>=</mo><mfrac><mn>7</mn><mrow><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup></mrow></mfrac><mo>=</mo><mfrac><mn>5</mn><mrow><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup></mrow></mfrac><mo>=</mo><mfrac><msub><mi>C</mi><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>C</mi></mrow></msub><mn>55</mn></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><mo>+</mo><mn>7</mn><mo>+</mo><mn>5</mn></mrow><mn>55</mn></mfrac><mo>=</mo><mfrac><mn>15</mn><mn>55</mn></mfrac><mo>=</mo><mfrac><mn>3</mn><mn>11</mn></mfrac></mstyle></math> (với <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><msub><mi>C</mi><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>C</mi></mrow></msub></mstyle></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><msub><mi>C</mi><mrow><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></mrow></msub></mstyle></math> lần lượt là chu vi của hai tam giác <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi>A</mi><mi>B</mi><mi>C</mi><mo>,</mo><msup><mi>A</mi><mo>'</mo></msup><msup><mi>B</mi><mo>'</mo></msup><msup><mi>C</mi><mo>'</mo></msup><mo>)</mo></mstyle></math><br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>+</mo><mo>)</mo><mfrac><mn>3</mn><mrow><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>B</mi><mo>'</mo></msup></mrow></mfrac><mo>=</mo><mfrac><mn>3</mn><mn>11</mn></mfrac><mo>⇒</mo><msup><mi>A</mi><mo>'</mo></msup><msup><mi>B</mi><mo>'</mo></msup><mo>=</mo><mfrac><mn>3.11</mn><mn>3</mn></mfrac><mo>=</mo><mpadded><mn>11</mn></mpadded><mi>cm</mi></mstyle></math><br />+)<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mrow><mfrac><mn>7</mn><mrow><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup></mrow></mfrac><mo>=</mo><mfrac><mn>3</mn><mn>11</mn></mfrac><mo>⇒</mo><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup><mo>=</mo><mfrac><mn>7.11</mn><mn>3</mn></mfrac><mo>≈</mo><mn>25</mn></mrow><mo>,</mo><mpadded><mn>67</mn></mpadded><mo>⁢</mo><mi>cm</mi></mstyle></math><br />+) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mrow><mfrac><mn>5</mn><mrow><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup></mrow></mfrac><mo>=</mo><mfrac><mn>3</mn><mn>11</mn></mfrac><mo>⇒</mo><msup><mi>A</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup><mo>=</mo><mfrac><mn>5.11</mn><mn>3</mn></mfrac><mo>≈</mo><mn>18</mn></mrow><mo>,</mo><mpadded><mn>33</mn></mpadded><mo>⁢</mo><mi>cm</mi></mstyle></math></p>
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