Bài 9. Ứng dụng thực tế của tam giác đồng dạng
Hướng dẫn giải Bài 55 (Trang 87 SGK Toán Hình học 8, Tập 2)
<p><strong class="content_question">Đề bài</strong></p>
<p>Hình 58 dưới đây mô tả dụng cụ đo bề dày của một số loại sản phẩm. Dụng cụ này gồm thước <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>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">C</span></span></span></span></span>được chia đến <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"><mn>1</mn></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-mn"><span class="mjx-char MJXc-TeX-main-R">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span> mm và gắn với một kim loại hình 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"><mi>A</mi><mi>B</mi><mi>D</mi></math>"><span id="MJXc-Node-8" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9" class="mjx-mrow"><span id="MJXc-Node-10" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-11" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span><span id="MJXc-Node-12" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">D</span></span></span></span></span>, khoảng cách <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><mo>=</mo><mn>10</mn></math>"><span id="MJXc-Node-13" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-14" class="mjx-mrow"><span id="MJXc-Node-15" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</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">10</span></span></span></span></span>.</p>
<p><img src="https://img.loigiaihay.com/picture/2018/0718/b55-trang-87-sgk-toan-8-t2-c2.jpg" /></p>
<p>Muốn đo bề dầy của vật, ta kẹp vật vào giữa bản kim loại và thước (đáy của vật áp vào bề mặt thước <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>C</mi></math>"><span id="MJXc-Node-19" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-20" class="mjx-mrow"><span id="MJXc-Node-21" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-22" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span></span></span></span>), khi đó trên thước <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>C</mi></math>"><span id="MJXc-Node-23" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-24" class="mjx-mrow"><span id="MJXc-Node-25" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-26" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span></span></span></span> ta đọc được bề dày <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>d</mi></math>"><span id="MJXc-Node-27" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-28" class="mjx-mrow"><span id="MJXc-Node-29" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">d</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></span></span> của vật (trên hình vẽ ta có <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"><mi>d</mi><mo>=</mo><mn>5</mn><mo>,</mo><mn>5</mn><mi>m</mi><mi>m</mi></math>"><span id="MJXc-Node-30" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-31" class="mjx-mrow"><span id="MJXc-Node-32" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">d</span></span><span id="MJXc-Node-33" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">=</span></span><span id="MJXc-Node-34" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">5</span></span><span id="MJXc-Node-35" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-36" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R">5</span></span><span id="MJXc-Node-37" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">m</span></span><span id="MJXc-Node-38" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">m</span></span></span></span></span>).</p>
<p>Hãy chỉ rõ định lí nào của hình học là cơ sở để ghi vạch trên thước <span id="MathJax-Element-9-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>C</mi></math>"><span id="MJXc-Node-39" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-40" class="mjx-mrow"><span id="MJXc-Node-41" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-42" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span></span></span></span> (<span id="MathJax-Element-10-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><mo>&#x2264;</mo><mn>10</mn></math>"><span id="MJXc-Node-43" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-44" class="mjx-mrow"><span id="MJXc-Node-45" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">d</span></span><span id="MJXc-Node-46" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">≤</span></span><span id="MJXc-Node-47" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">10mm</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/2019/0315/h3-bai-55-trang-87-sgk-toan-8-t2.jpg" /></strong></p>
<p><span class="content_detail">Theo hình vẽ thì <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>B</mi><mo>'</mo></msup><msup><mi>C</mi><mo>'</mo></msup><mo>/</mo><mo>/</mo><mi>B</mi><mi>C</mi></math>$ nên <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><mo>⁢</mo><mo>~</mo><mo>⁢</mo><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> (theo định lí)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mi>A</mi><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup></mrow><mrow><mi>A</mi><mo>⁢</mo><mi>C</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></math><br /><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></mstyle></math> là bề dày của vật cần đo<br />Vì <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi>d</mi><mo>≤</mo><mpadded><mn>10</mn></mpadded><mo>⁢</mo><mi>mm</mi></mstyle></math> nên <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi>B</mi><mo>⁢</mo><mi>C</mi><mo>=</mo><mpadded><mn>10</mn></mpadded><mo>⁢</mo><mi>mm</mi><mo>=</mo><mpadded><mn>1</mn></mpadded><mo>⁢</mo><mi>cm</mi></mstyle></math><br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><msup><mi>B</mi><mo>'</mo></msup><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup><mo>=</mo><mfrac><mrow><mi>A</mi><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup><mo>⋅</mo><mi>B</mi><mo>⁢</mo><mi>C</mi></mrow><mrow><mi>A</mi><mo>⁢</mo><mi>C</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>10</mn></mfrac><mo>⁢</mo><mi>A</mi><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup></mstyle></math><br />Vậy khi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi>A</mi><mo>⁢</mo><msup><mi>C</mi><mo>'</mo></msup><mo>=</mo><mn>5</mn><mo>,</mo><mpadded><mn>5</mn></mpadded><mo>⁢</mo><mi>cm</mi></mstyle></math> thì đọc <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><mfrac><mn>1</mn><mn>10</mn></mfrac><mo>⁢</mo><mn>.5</mn><mo>,</mo><mpadded><mn>5</mn></mpadded><mo>⁢</mo><mi>cm</mi></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>,</mo><mpadded><mn>55</mn></mpadded><mo>⁢</mo><mi>cm</mi><mo>=</mo><mn>5</mn><mo>,</mo><mpadded><mn>5</mn></mpadded><mo>⁢</mo><mi>mm</mi><mo>.</mo></math><br /></span><span class="content_detail">Dụng cụ trên đã dùng tính chất hai tam giác đồng dạng thì cạnh tương ứng tỉ lệ.</span><strong class="content_detail"><br /></strong></p>
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