Bài 3: Tính Chất Đường Phân Giác Của Tam Giác
Hướng dẫn giải Bài 20 (Trang 68 SGK Toán Hình học 8, Tập 2)
<p><strong class="content_question">Đề bài</strong></p>
<p>Cho hình thang <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><mi>D</mi><mspace width="thickmathspace" /><mo stretchy="false">(</mo><mi>A</mi><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>C</mi><mi>D</mi><mo stretchy="false">)</mo></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 id="MJXc-Node-6" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">D</span></span><span id="MJXc-Node-7" class="mjx-mspace"></span><span id="MJXc-Node-8" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">(</span></span><span id="MJXc-Node-9" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-10" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span><span id="MJXc-Node-11" class="mjx-texatom"><span id="MJXc-Node-12" class="mjx-mrow"><span id="MJXc-Node-13" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">/</span></span></span></span><span id="MJXc-Node-14" class="mjx-texatom"><span id="MJXc-Node-15" class="mjx-mrow"><span id="MJXc-Node-16" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">/</span></span></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">D</span></span><span id="MJXc-Node-19" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">)</span></span></span></span></span>. Hai đường chéo <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>C</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">A</span></span><span id="MJXc-Node-23" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span></span></span></span> và <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>D</mi></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>D</mi></math></span></span> cắt nhau tại <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>O</mi></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">O</span></span></span></span></span>. Đường thẳng <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></math>"><span id="MJXc-Node-31" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-32" class="mjx-mrow"><span id="MJXc-Node-33" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">a</span></span></span></span></span> qua <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>O</mi></math>"><span id="MJXc-Node-34" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-35" class="mjx-mrow"><span id="MJXc-Node-36" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">O</span></span></span></span></span> và song song với đáy của hình thang cắt các cạnh <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>D</mi><mo>,</mo><mi>B</mi><mi>C</mi></math>"><span id="MJXc-Node-37" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-38" class="mjx-mrow"><span id="MJXc-Node-39" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-40" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">D</span></span><span id="MJXc-Node-41" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-42" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I">B</span></span><span id="MJXc-Node-43" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span></span></span></span> theo thứ tự <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>E</mi></math>"><span id="MJXc-Node-44" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-45" class="mjx-mrow"><span id="MJXc-Node-46" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">E</span></span></span></span></span> và <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>F</mi></math>"><span id="MJXc-Node-47" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-48" class="mjx-mrow"><span id="MJXc-Node-49" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">F</span></span></span></span></span> (h26)</p>
<p>Chứng minh rằng OE=OF.</p>
<p><img src="https://img.loigiaihay.com/picture/2018/0718/b20-trang-68-sgk-toan-8-t2-c2.jpg" /></p>
<p><strong class="content_detail">Lời giải chi tiết</strong></p>
<p><span class="content_detail"><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>D</mi><mo>⁢</mo><mi>C</mi></mstyle></math> có OE//DC (gt) nên <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mfrac><mrow><mi>O</mi><mo>⁢</mo><mi>E</mi></mrow><mrow><mi>D</mi><mo>⁢</mo><mi>C</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>A</mi><mo>⁢</mo><mi>O</mi></mrow><mrow><mi>A</mi><mo>⁢</mo><mi>C</mi></mrow></mfrac></mstyle></math> (1) (hệ quả của định lí TaLet trong tam giác)</span></p>
<p><span class="content_detail"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>D</mi><mo>⁢</mo><mi>C</mi></mstyle></math> có OF//DC (gt) nên <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mfrac><mrow><mi>O</mi><mo>⁢</mo><mi>F</mi></mrow><mrow><mi>D</mi><mo>⁢</mo><mi>C</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>B</mi><mo>⁢</mo><mi>F</mi></mrow><mrow><mi>B</mi><mo>⁢</mo><mi>C</mi></mrow></mfrac></mstyle></math> (2) (hệ quả của định lí TaLet trong tam giác) </span></p>
<p><span class="content_detail"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi mathvariant="normal">△</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>A</mi><mo>⁢</mo><mi>C</mi></mstyle></math> có OF//AB (gt) nên <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mfrac><mrow><mi>A</mi><mo>⁢</mo><mi>O</mi></mrow><mrow><mi>A</mi><mo>⁢</mo><mi>C</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>B</mi><mo>⁢</mo><mi>F</mi></mrow><mrow><mi>B</mi><mo>⁢</mo><mi>C</mi></mrow></mfrac></mstyle></math> (3) (hệ quả của định lí TaLet trong tam giác) </span></p>
<p><span class="content_detail">Từ (1), (2), (3) suy ra <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mfrac><mrow><mi>O</mi><mo>⁢</mo><mi>E</mi></mrow><mrow><mi>D</mi><mo>⁢</mo><mi>C</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>O</mi><mo>⁢</mo><mi>F</mi></mrow><mrow><mi>D</mi><mo>⁢</mo><mi>C</mi></mrow></mfrac></mstyle></math> nên OE=OF.<br /></span></p>
<p> </p>
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