Bài 9: Hình Chữ Nhật
Hướng dẫn giải Bài 66 (Trang 100 SGK Toán Hình học 8, Tập 1)
<p><strong class="content_question">Đề bài</strong></p>
<p>Đố. Một đội công nhân đang trồng cây trên đoạn đường <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></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></span></span> thì gặp chướng ngại vật che lấp tầm nhìn (h.<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>92</mn></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>92</mn></math></span></span>). Đội đã dựng các điểm <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>C</mi><mo>,</mo><mi>D</mi><mo>,</mo><mi>E</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">C</span></span><span id="MJXc-Node-11" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-12" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I">D</span></span><span id="MJXc-Node-13" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-14" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I">E</span></span></span></span></span> như trên hình vẽ rồi trồng cây tiếp trên đoạn thẳng <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>E</mi><mi>F</mi></math>"><span id="MJXc-Node-15" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-16" class="mjx-mrow"><span id="MJXc-Node-17" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">E</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math></span></span>vuông góc với <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>D</mi><mi>E</mi></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mi>E</mi></math></span></span>. Vì sao <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>B</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></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span> và <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>E</mi><mi>F</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">E</span></span><span id="MJXc-Node-30" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">F</span></span></span></span></span> cùng nằm trên một đường thẳng ?</p>
<p><strong class="content_detail">Lời giải chi tiết</strong></p>
<p>Tứ giác <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>B</mi><mi>C</mi><mi>D</mi><mi>E</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">B</span></span><span id="MJXc-Node-34" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span><span id="MJXc-Node-35" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">D</span></span><span id="MJXc-Node-36" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">E</span></span></span></span></span> có:</p>
<p><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>B</mi><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>D</mi><mi>E</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">B</span></span><span id="MJXc-Node-40" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span><span id="MJXc-Node-41" class="mjx-texatom"><span id="MJXc-Node-42" class="mjx-mrow"><span id="MJXc-Node-43" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">/</span></span></span></span><span id="MJXc-Node-44" class="mjx-texatom"><span id="MJXc-Node-45" class="mjx-mrow"><span id="MJXc-Node-46" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">/</span></span></span></span><span id="MJXc-Node-47" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">D</span></span><span id="MJXc-Node-48" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">E</span></span></span></span></span> (vì cùng vuông góc với <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>C</mi><mi>D</mi></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-mi"><span class="mjx-char MJXc-TeX-math-I">C</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>)</p>
<p><span id="MathJax-Element-11-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><mi>D</mi><mi>E</mi></math>"><span id="MJXc-Node-53" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-54" class="mjx-mrow"><span id="MJXc-Node-55" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span><span id="MJXc-Node-56" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span><span id="MJXc-Node-57" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">=</span></span><span id="MJXc-Node-58" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I">D</span></span><span id="MJXc-Node-59" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">E</span></span></span></span></span> (giả thiết)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo></mstyle></math> Tứ giác BCDE là hình bình hành (dấu hiệu nhận biết hình<br />bình hành)<br />Mà <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>B</mi><mo>⁢</mo><mi>C</mi><mo>⁢</mo><mi>D</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>90</mn><mn>0</mn></msup></mstyle></math> (giả thiết)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo></mstyle></math> Hình bình hành BCDE là hình chữ nhật (dấu hiệu nhận biết<br />hình chữ nhật)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mover accent="true"><mrow><mi>C</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>E</mi></mrow><mo>^</mo></mover><mo>=</mo><mover accent="true"><mrow><mi>B</mi><mo>⁢</mo><mi>E</mi><mo>⁢</mo><mi>D</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>90</mn><mn>0</mn></msup></mstyle></math><br />Mặt khác: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>C</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>A</mi></mrow><mo>^</mo></mover><mo>=</mo><mover accent="true"><mrow><mi>F</mi><mo>⁢</mo><mi>E</mi><mo>⁢</mo><mi>D</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>90</mn><mn>0</mn></msup></mstyle></math> (giả thiết)<br />Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>C</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>A</mi></mrow><mo>^</mo></mover><mo>+</mo><mover accent="true"><mrow><mi>C</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>E</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>90</mn><mn>0</mn></msup><mo>+</mo><msup><mn>90</mn><mn>0</mn></msup><mo>=</mo><msup><mn>180</mn><mn>0</mn></msup></mstyle></math><br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>E</mi></mstyle></math> thẳng hàng (1)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>F</mi><mo>⁢</mo><mi>E</mi><mo>⁢</mo><mi>D</mi></mrow><mo>^</mo></mover><mo>+</mo><mover accent="true"><mrow><mi>B</mi><mo>⁢</mo><mi>E</mi><mo>⁢</mo><mi>D</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>90</mn><mn>0</mn></msup><mo>+</mo><msup><mn>90</mn><mn>0</mn></msup><mo>=</mo><msup><mn>180</mn><mn>0</mn></msup></mstyle></math><br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mi>B</mi><mo>,</mo><mi>E</mi><mo>,</mo><mi>F</mi></mstyle></math> thẳng hàng (2)<br />Từ (1) và (2) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>A</mi><mo>⁢</mo><mi>B</mi><mo> </mo><mi>v</mi><mi>à</mi><mo>⁢</mo><mi>E</mi><mi>F</mi></math> cùng nằm trên một đường thẳng.<br />Cách khác:<br />Tứ giác BCDE có:<br />BC//DE (vì cùng vuông góc với CD)<br />BC=DE (giả thiết)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo></mstyle></math> Tứ giác BCDE là hình bình hành (dấu hiệu nhận biết hình bình hành)<br />Suy ra <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>E</mi><mo>/</mo><mo>/</mo><mi>C</mi><mi>D</mi><mrow><mo>(</mo><mmultiscripts><mo>)</mo><mprescripts/><none/><mo>*</mo></mmultiscripts></mrow></math> (tính chất)<br />Lại có <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>C</mi></mrow><mo>^</mo></mover><mo>=</mo><mover accent="true"><mrow><mi>B</mi><mo>⁢</mo><mi>C</mi><mo>⁢</mo><mi>D</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>90</mn><mn>0</mn></msup></mstyle></math> mà hai góc này ở vị trí so le trong nên AB//CD(* *)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>D</mi><mo>⁢</mo><mi>E</mi><mo>⁢</mo><mi>F</mi></mrow><mo>^</mo></mover><mo>=</mo><mover accent="true"><mrow><mi>C</mi><mo>⁢</mo><mi>D</mi><mo>⁢</mo><mi>E</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>90</mn><mn>0</mn></msup></mstyle></math> mà hai góc này ở vị trí so le trong nên EF//CD(* * *)<br />Từ' <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mrow><mo>(</mo><mmultiscripts><mo>)</mo><mprescripts/><none/><mo>*</mo></mmultiscripts></mrow><mo>,</mo><mrow><mo>(</mo><mmultiscripts><mo>)</mo><mprescripts/><none/><mrow><mo>*</mo><mo>*</mo></mrow></mmultiscripts></mrow><mo>,</mo><mrow><mo>(</mo><mmultiscripts><mo>)</mo><mprescripts/><none/><mrow><mo>*</mo><mo>*</mo></mrow></mmultiscripts></mrow></mstyle></math> và theo tiên đề ơclit ta suy ra A, B, E, F thẳng hàng<br />Hay AB và EF cùng nằm trên một đường thẳng.</p>
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<p> </p>
<p> </p>
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