Ôn Tập Chương 1
Hướng dẫn giải Bài 88 (Trang 111 SGK Toán Hình học 8, Tập 1)
<p><strong class="content_question">Đề bài</strong></p>
<p>Cho tứ 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><mi>D</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 id="MJXc-Node-6" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">D</span></span></span></span></span>. Gọi <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>E</mi><mo>,</mo><mi>F</mi><mo>,</mo><mi>G</mi><mo>,</mo><mi>H</mi></math>"><span id="MJXc-Node-7" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8" class="mjx-mrow"><span id="MJXc-Node-9" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">E</span></span><span id="MJXc-Node-10" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-11" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I">F</span></span><span id="MJXc-Node-12" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-13" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I">G</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">H</span></span></span></span></span> theo thứ tự là trung điểm của <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><mo>,</mo><mi>B</mi><mi>C</mi><mo>,</mo><mi>C</mi><mi>D</mi><mo>,</mo><mi>D</mi><mi>A</mi><mo>.</mo></math>"><span id="MJXc-Node-16" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-17" class="mjx-mrow"><span id="MJXc-Node-18" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-19" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span><span id="MJXc-Node-20" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-21" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I">B</span></span><span id="MJXc-Node-22" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span><span id="MJXc-Node-23" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-24" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I">C</span></span><span id="MJXc-Node-25" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">D</span></span><span id="MJXc-Node-26" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-27" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I">D</span></span><span id="MJXc-Node-28" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-29" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">.</span></span></span></span></span> Các đường chéo <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>C</mi><mo>,</mo><mi>B</mi><mi>D</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">A</span></span><span id="MJXc-Node-33" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span><span id="MJXc-Node-34" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-35" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I">B</span></span><span id="MJXc-Node-36" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">D</span></span></span></span></span>của tứ giác <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>B</mi><mi>C</mi><mi>D</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">B</span></span><span id="MJXc-Node-41" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span><span id="MJXc-Node-42" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">D</span></span></span></span></span> có điều kiện gì thì <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>E</mi><mi>F</mi><mi>G</mi><mi>H</mi></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">E</span></span><span id="MJXc-Node-46" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">F</span></span><span id="MJXc-Node-47" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">G</span></span><span id="MJXc-Node-48" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">H</span></span></span></span></span> là:</p>
<p>a) Hình chữ nhật?</p>
<p>b) Hình thoi? </p>
<p>c) Hình vuông</p>
<p><strong class="content_detail">Lời giải chi tiết</strong></p>
<p><img src="https://img.loigiaihay.com/picture/2018/0713/b88-trang-111-sgk-toan-8-t-1-c2.jpg" alt="" /></p>
<p>+ Ta có: EB=EA, FB=FC (gt)<br />Do đó EF là đường trung bình của tam giác ABC<br />Suy ra <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>F</mi><mo>/</mo><mo>/</mo><mi>A</mi><mi>C</mi><mo>,</mo><mi>E</mi><mi>F</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>A</mi><mi>C</mi></math> (tính chất đường trung bình của tam giác)<br />+ Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi>H</mi><mo>⁢</mo><mi>D</mi><mo>=</mo><mi>H</mi><mo>⁢</mo><mi>A</mi><mo>,</mo><mi>G</mi><mo>⁢</mo><mi>D</mi><mo>=</mo><mi>G</mi><mo>⁢</mo><mi>C</mi><mo>⁢</mo><mrow><mo>(</mo><mi>gt</mi><mo>)</mo></mrow></mstyle></math><br />Do đó HG là đường trung bình của tam giác ADC<br />Suy ra <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi>H</mi><mi>G</mi><mo>/</mo><mo>/</mo><mi>A</mi><mi>C</mi><mo>,</mo><mi>H</mi><mi>G</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>A</mi><mi>C</mi></mstyle></math> (tính chất đường trung bình của tam giác)</p>
<p>Do đó EF//HG, EF=HG nên EFGH là hình bình hành.</p>
<p>+ Ta có: EB=EA, AH=HD (gt)<br />Do đó EH là đường trung bình của tam giác ABD.<br />Suy ra <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi>E</mi><mi>H</mi><mo>/</mo><mo>/</mo><mi>B</mi><mi>D</mi><mo>,</mo><mi>E</mi><mi>H</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>B</mi><mi>D</mi></mstyle></math> (tính chất đường trung bình của tam giác)<br />a) Hình bình hành EFGH là hình chữ nhật <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇔</mo><mi>E</mi><mo>⁢</mo><mi>H</mi><mo>⟂</mo><mi>E</mi><mo>⁢</mo><mi>F</mi></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇔</mo><mi>A</mi><mo>⁢</mo><mi>C</mi><mo>⟂</mo><mi>B</mi><mo>⁢</mo><mi>D</mi></mstyle></math> (vì EH//BD; EF//AC)</p>
<p>Điều kiện phải tìm: các đường chéo AC và BD vuông góc với nhau.<br />b) Hình bình hành EFGH là hình thoi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇔</mo><mi>E</mi><mo>⁢</mo><mi>F</mi><mo>=</mo><mi>E</mi><mo>⁢</mo><mi>H</mi></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇔</mo><mi>A</mi><mo>⁢</mo><mi>C</mi><mo>=</mo><mi>B</mi><mo>⁢</mo><mi>D</mi></mstyle></math> (vì <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><mi>A</mi><mo>⁢</mo><mi>C</mi><mo>,</mo><mi>E</mi><mo>⁢</mo><mi>H</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>D</mi></math>)<br />Điều kiện phải tìm: các đường chéo AC và BD bằng nhau.</p>
<p>c) Hình bình hành EFGH là hình vuông khi và chỉ khi<br />EFGH vừa là hình chữ nhật đồng thời là hình thoi. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mi>A</mi><mo>⁢</mo><mi>C</mi><mo>⟂</mo><mi>B</mi><mo>⁢</mo><mi>D</mi></mstyle></math> và AC=BD.<br />Điều kiện phải tìm: các đường chéo AC, BD bằng nhau và vuông góc với nhau.<br /><br /><br /></p>
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