Bài 8. Các trường hợp đồng dạng của tam giác vuông
Hướng dẫn giải Bài 51 (Trang 84 SGK Toán Hình học 8, Tập 2)
<p><strong class="content_question">Đề bài</strong></p>
<p>Chân đường cao <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>H</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">H</span></span></span></span></span> của tam giác vuông <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 id="MJXc-Node-5" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6" class="mjx-mrow"><span id="MJXc-Node-7" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-8" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span><span id="MJXc-Node-9" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span></span></span></span> chia cạnh huyền <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>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> thành hai đoạn có độ dài <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"><mn>25</mn><mi>c</mi><mi>m</mi></math>"><span id="MJXc-Node-14" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-15" class="mjx-mrow"><span id="MJXc-Node-16" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R">25</span></span><span id="MJXc-Node-17" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">c</span></span><span id="MJXc-Node-18" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">m</span></span></span></span></span>và <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>36</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>36</mn><mi>c</mi><mi>m</mi></math></span></span>. Tính chu vi và diện tích của tam giác vuông đó (h.53)</p>
<p>Hướng dẫn: Trước tiên tìm cách <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 class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>H</mi></math></span></span> từ các tam giác vuông đồng dạng, sau đó tính các cạnh của tam giác <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"><mi>A</mi><mi>B</mi><mi>C</mi><mo>.</mo></math>"><span id="MJXc-Node-28" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-29" class="mjx-mrow"><span id="MJXc-Node-30" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-31" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span><span id="MJXc-Node-32" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span><span id="MJXc-Node-33" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">.</span></span></span></span></span></p>
<p> </p>
<p><strong class="content_question"><img class="wscnph" style="max-width: 100%;" src="https://static.colearn.vn:8413/v1.0/upload/library/25072022/b51-trand-84-sdk-toan-8-t2-c2-XadHzl.jpg" /></strong></p>
<p><strong class="content_question">Lời giải chi tiết<br /></strong><span class="content_question"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi mathvariant="normal">△</mi><mo>⁢</mo><mi>A</mi><mo>⁢</mo><mi>H</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mo>~</mo><mo>⁢</mo><mi mathvariant="normal">△</mi><mo>⁢</mo><mi>C</mi><mo>⁢</mo><mi>H</mi><mo>⁢</mo><mi>A</mi><mo mathvariant="italic"> </mo></mstyle></math> (g.g) $\quad$ vì <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mpadded lspace="+10px"><mover accent="true"><mrow><mi>A</mi><mo>⁢</mo><mi>H</mi><mo>⁢</mo><mi>B</mi></mrow><mo>^</mo></mover></mpadded><mo>=</mo><mover accent="true"><mrow><mi>A</mi><mo>⁢</mo><mi>H</mi><mo>⁢</mo><mi>C</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>90</mn><mo>∘</mo></msup></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>B</mi><mo>⁢</mo><mi>A</mi><mo>⁢</mo><mi>H</mi></mrow><mo>^</mo></mover><mo>=</mo><mover accent="true"><mrow><mi>A</mi><mo>⁢</mo><mi>C</mi><mo>⁢</mo><mi>H</mi></mrow><mo>^</mo></mover></mstyle></math> (cùng phụ với <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>H</mi><mo>⁢</mo><mi>A</mi><mo>⁢</mo><mi>C</mi></mrow><mo>^</mo></mover></mstyle></math>)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>⇒</mo><mfrac><mrow><mi>A</mi><mo>⁢</mo><mi>H</mi></mrow><mrow><mi>C</mi><mo>⁢</mo><mi>H</mi></mrow></mfrac><mo>=</mo><mpadded><mfrac><mrow><mi>B</mi><mo>⁢</mo><mi>H</mi></mrow><mrow><mi>A</mi><mo>⁢</mo><mi>H</mi></mrow></mfrac></mpadded><mo>⁢</mo></mrow><mspace linebreak="newline"/><mo>⇒</mo><mi>A</mi><mo>⁢</mo><msup><mi>H</mi><mn>2</mn></msup><mo>=</mo><mi>C</mi><mo>⁢</mo><mi>H</mi><mo>.</mo><mi>B</mi><mo>⁢</mo><mi>H</mi><mo>=</mo><mpadded><mn>25.36</mn></mpadded><mspace linebreak="newline"/><mo>⇒</mo><mi>A</mi><mo>⁢</mo><msup><mi>H</mi><mn>2</mn></msup><mo>=</mo><mn>900</mn><mo>⇒</mo><mi>A</mi><mo>⁢</mo><mi>H</mi><mo>=</mo><mpadded><mn>30</mn></mpadded><mo>⁢</mo><mpadded><mi>cm</mi></mpadded></math><br /></span></p>
<p><span class="content_question">Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>C</mi></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⋅</mo><mi>A</mi><mo>⁢</mo><mi>H</mi><mo>.</mo><mi>B</mi><mo>⁢</mo><mi>C</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⋅</mo><mn>30</mn><mo>.</mo><mrow><mo>(</mo><mn>25</mn><mo>+</mo><mn>36</mn><mo>)</mo></mrow><mo>=</mo><mpadded><mn>915</mn></mpadded><mo>⁢</mo><msup><mi>cm</mi><mn>2</mn></msup></math></span></p>
<p><span class="content_question">Áp dụng Py-ta-go cho 2 tam giác vuông ABH và ACH ta được:<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><msup><mi>B</mi><mn>2</mn></msup><mo>=</mo><mi>B</mi><msup><mi>H</mi><mn>2</mn></msup><mo>+</mo><mi>A</mi><mpadded><msup><mi>H</mi><mn>2</mn></msup></mpadded><mi mathvariant="normal">&</mi><mspace linebreak="newline"/><mo>⇒</mo><mi>A</mi><msup><mi>B</mi><mn>2</mn></msup><mo>=</mo><msup><mn>25</mn><mn>2</mn></msup><mo>+</mo><msup><mn>30</mn><mn>2</mn></msup><mo>=</mo><mpadded><mn>1525</mn></mpadded><mspace linebreak="newline"/><mo>⇒</mo><mi>A</mi><mi>B</mi><mo>≈</mo><mn>39</mn><mo>,</mo><mpadded><mn>05</mn></mpadded><mpadded><mi>cm</mi></mpadded><mspace linebreak="newline"/><mi>A</mi><msup><mi>C</mi><mn>2</mn></msup><mo>=</mo><mi>C</mi><msup><mi>H</mi><mn>2</mn></msup><mo>+</mo><mi>A</mi><mpadded><msup><mi>H</mi><mn>2</mn></msup></mpadded><mspace linebreak="newline"/><mo>⇒</mo><mi>A</mi><msup><mi>C</mi><mn>2</mn></msup><mo>=</mo><msup><mn>36</mn><mn>2</mn></msup><mo>+</mo><msup><mn>30</mn><mn>2</mn></msup><mo>=</mo><mpadded><mn>2196</mn></mpadded><mspace linebreak="newline"/><mo>⇒</mo><mi>A</mi><mi>C</mi><mo>≈</mo><mn>46</mn><mo>,</mo><mpadded><mn>86</mn></mpadded><mpadded><mi>cm</mi></mpadded></math><br /></span></p>
<p><span class="content_question">Chu vi tam giác ABC là: <span id="MathJax-Element-20-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: 400; font-size: 16.94px; letter-spacing: normal; overflow-wrap: 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: #000000; font-family: OpenSans, Tahoma, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; 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>P</mi><mo>=</mo><mi>A</mi><mi>B</mi><mo>+</mo><mi>A</mi><mi>C</mi><mo>+</mo><mi>B</mi><mi>C</mi></math>"><span id="MJXc-Node-324" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-325" class="mjx-mrow"><span id="MJXc-Node-326" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">P</span></span><span id="MJXc-Node-327" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">=</span></span><span id="MJXc-Node-328" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-329" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span><span id="MJXc-Node-330" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R">+</span></span><span id="MJXc-Node-331" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-332" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span><span id="MJXc-Node-333" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R">+</span></span><span id="MJXc-Node-334" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I">B</span></span><span id="MJXc-Node-335" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span></span></span></span> <span id="MathJax-Element-21-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: 400; font-size: 16.94px; letter-spacing: normal; overflow-wrap: 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: #000000; font-family: OpenSans, Tahoma, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; 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"><mo>=</mo><mn>39</mn><mo>,</mo><mn>05</mn><mo>+</mo><mn>46</mn><mo>,</mo><mn>86</mn><mo>+</mo><mn>61</mn><mo>=</mo><mn>146</mn><mo>,</mo><mn>91</mn><mi>c</mi><mi>m</mi></math>"><span id="MJXc-Node-336" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-337" class="mjx-mrow"><span id="MJXc-Node-338" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">=</span></span><span id="MJXc-Node-339" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">39</span></span><span id="MJXc-Node-340" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-341" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R">05</span></span><span id="MJXc-Node-342" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R">+</span></span><span id="MJXc-Node-343" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R">46</span></span><span id="MJXc-Node-344" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-345" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R">86</span></span><span id="MJXc-Node-346" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R">+</span></span><span id="MJXc-Node-347" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R">61</span></span><span id="MJXc-Node-348" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">=</span></span><span id="MJXc-Node-349" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">146</span></span><span id="MJXc-Node-350" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-351" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R">91</span></span><span id="MJXc-Node-352" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">c</span></span><span id="MJXc-Node-353" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">m</span></span></span></span></span><br /><br /></span></p>
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