Bài 9: Hình Chữ Nhật
Hướng dẫn giải Bài 64 (Trang 100 SGK Toán Hình học 8, Tập 1)
<p><strong class="content_question">Đề bài</strong></p>
<p>Cho hình bình hành <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>. Các tia phân giác của các góc <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><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>,</mo><mi>D</mi></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>,</mo><mi>D</mi></math></span></span> cắt nhau như trên hình <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"><mn>91.</mn></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-mn"><span class="mjx-char MJXc-TeX-main-R">91.</span></span></span></span></span> Chứng minh rằng <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><mi>G</mi><mi>H</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">E</span></span><span id="MJXc-Node-22" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">F</span></span><span id="MJXc-Node-23" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">G</span></span><span id="MJXc-Node-24" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">H</span></span></span></span></span> là hình chữ nhật. </p>
<p><img class="wscnph" style="max-width: 100%;" src="https://static.colearn.vn:8413/v1.0/upload/library/05072022/b64-trand-100-sdk-toan-8-t-1-c2-ZOA535.jpg" /></p>
<p><strong>Lời giải chi tiết</strong></p>
<p>Theo giả thiết ABCD là hình bình hành nên AD//BC, AB//CD<br />Vì <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>D</mi><mo>/</mo><mo>/</mo><mi>B</mi><mi>C</mi><mo>⇒</mo><mover accent="true"><mrow><mi>D</mi><mo></mo><mi>A</mi><mo></mo><mi>B</mi></mrow><mo>^</mo></mover><mo>+</mo><mover accent="true"><mrow><mi>A</mi><mo></mo><mi>B</mi><mo></mo><mi>C</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>180</mn><mn>0</mn></msup></math> (hai góc trong cùng phía bù nhau)<br />Vì AG là tia phân giác <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>D</mi><mo></mo><mi>A</mi><mo></mo><mi>B</mi></mrow><mo>^</mo></mover></mstyle></math> (giả thiết)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mover accent="true"><mrow><mi>B</mi><mo></mo><mi>A</mi><mo></mo><mi>G</mi></mrow><mo>^</mo></mover><mo>=</mo><mover accent="true"><mrow><mi>D</mi><mo></mo><mi>A</mi><mo></mo><mi>H</mi></mrow><mo>^</mo></mover><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mover accent="true"><mrow><mi>D</mi><mo></mo><mi>A</mi><mo></mo><mi>B</mi></mrow><mo>^</mo></mover></mstyle></math> (tính chất tia phân giác)<br />Vì BG là tia phân giác <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></mstyle></math> (giả thiết)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mover accent="true"><mrow><mi>A</mi><mo></mo><mi>B</mi><mo></mo><mi>G</mi></mrow><mo>^</mo></mover><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mover accent="true"><mrow><mi>A</mi><mo></mo><mi>B</mi><mo></mo><mi>C</mi></mrow><mo>^</mo></mover><mo></mo></math><br />Do đó: <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>G</mi></mrow><mo>^</mo></mover><mo>+</mo><mover accent="true"><mrow><mi>A</mi><mo></mo><mi>B</mi><mo></mo><mi>G</mi></mrow><mo>^</mo></mover><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo>(</mo><mover accent="true"><mrow><mi>D</mi><mo></mo><mi>A</mi><mo></mo><mi>B</mi></mrow><mo>^</mo></mover><mo>+</mo><mover accent="true"><mrow><mi>A</mi><mo></mo><mi>B</mi><mo></mo><mi>C</mi></mrow><mo>^</mo></mover><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⋅</mo><msup><mn>180</mn><mn>0</mn></msup><mo>=</mo><msup><mn>90</mn><mn>0</mn></msup></mstyle></math><br />Xét <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi mathvariant="normal">△</mi><mo></mo><mi>A</mi><mo></mo><mi>G</mi><mo></mo><mi>B</mi></mstyle></math> có:<br /><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>G</mi></mrow><mo>^</mo></mover><mo>+</mo><mover accent="true"><mrow><mi>A</mi><mo></mo><mi>B</mi><mo></mo><mi>G</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>90</mn><mn>0</mn></msup></mstyle></math><br />Áp dụng định lí tổng ba góc trong một tam giác vào tam giác AGB ta có:<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>B</mi><mo></mo><mi>A</mi><mo></mo><mi>G</mi></mrow><mo>^</mo></mover><mo>+</mo><mover accent="true"><mrow><mi>A</mi><mo></mo><mi>B</mi><mo></mo><mi>G</mi></mrow><mo>^</mo></mover><mo>+</mo><mover accent="true"><mrow><mi>A</mi><mo></mo><mi>G</mi><mo></mo><mi>B</mi></mrow><mo>^</mo></mover><mo>=</mo><mpadded><msup><mn>180</mn><mn>0</mn></msup></mpadded><mspace linebreak="newline"></mspace><mo>⇒</mo><mover accent="true"><mrow><mi>A</mi><mo></mo><mi>G</mi><mo></mo><mi>B</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>180</mn><mn>0</mn></msup><mo>-</mo><mrow><mo>(</mo><mover accent="true"><mrow><mi>B</mi><mo></mo><mi>A</mi><mo></mo><mi>G</mi></mrow><mo>^</mo></mover><mo>+</mo><mover accent="true"><mrow><mi>A</mi><mo></mo><mi>B</mi><mo></mo><mi>G</mi></mrow><mo>^</mo></mover><mo>)</mo></mrow><mo>=</mo><msup><mn>180</mn><mn>0</mn></msup><mo>-</mo><msup><mn>90</mn><mn>0</mn></msup><mo>=</mo><msup><mn>90</mn><mn>0</mn></msup><mo> </mo><mrow><mo>(</mo><mmultiscripts><mo>)</mo><mprescripts></mprescripts><mo>*</mo></mmultiscripts></mrow></math></p>
<p>+ Vì <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi>A</mi><mi>B</mi><mo>/</mo><mo>/</mo><mi>D</mi><mi>C</mi><mo>⇒</mo><mover accent="true"><mrow><mi>D</mi><mo></mo><mi>A</mi><mo></mo><mi>B</mi></mrow><mo>^</mo></mover><mo>+</mo><mover accent="true"><mrow><mi>A</mi><mo></mo><mi>D</mi><mo></mo><mi>C</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>180</mn><mn>0</mn></msup></mstyle></math> (hai góc trong cùng phía bù nhau)<br />+ Vì DE là tia phân giác <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>A</mi><mo></mo><mi>D</mi><mo></mo><mi>C</mi></mrow><mo>^</mo></mover></mstyle></math> (giả thiết)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mover accent="true"><mrow><mi>A</mi><mo></mo><mi>D</mi><mo></mo><mi>H</mi></mrow><mo>^</mo></mover><mo>=</mo><mover accent="true"><mrow><mi>E</mi><mo></mo><mi>D</mi><mo></mo><mi>C</mi></mrow><mo>^</mo></mover><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mover accent="true"><mrow><mi>A</mi><mo></mo><mi>D</mi><mo></mo><mi>C</mi></mrow><mo>^</mo></mover></mstyle></math> (tính chất tia phân giác)<br />Do đó: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>D</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>D</mi><mo></mo><mi>H</mi></mrow><mo>^</mo></mover><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo>(</mo><mover accent="true"><mrow><mi>D</mi><mo></mo><mi>A</mi><mo></mo><mi>B</mi></mrow><mo>^</mo></mover><mo>+</mo><mover accent="true"><mrow><mi>A</mi><mo></mo><mi>D</mi><mo></mo><mi>C</mi></mrow><mo>^</mo></mover><mo>)</mo></mrow></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⋅</mo><msup><mn>180</mn><mn>0</mn></msup><mo>=</mo><msup><mn>90</mn><mn>0</mn></msup></mstyle></math><br />Áp dụng định lí tổng ba góc trong một tam giác vào tam giác ADH ta có:<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>D</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>D</mi><mo></mo><mi>H</mi></mrow><mo>^</mo></mover><mo>+</mo><mover accent="true"><mrow><mi>A</mi><mo></mo><mi>H</mi><mo></mo><mi>D</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>180</mn><mn>0</mn></msup></mstyle></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mover accent="true"><mrow><mi>A</mi><mo></mo><mi>H</mi><mo></mo><mi>D</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>180</mn><mo>∘</mo></msup><mo>-</mo><mrow><mo>(</mo><mover accent="true"><mrow><mi>D</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>D</mi><mo></mo><mi>H</mi></mrow><mo>^</mo></mover><mo>)</mo></mrow><mo>=</mo><msup><mn>180</mn><mo>∘</mo></msup><mo>-</mo><msup><mn>90</mn><mo>∘</mo></msup><mo>=</mo><msup><mn>90</mn><mo>∘</mo></msup></mstyle></math><br />Suy ra <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>H</mi><mo>⟂</mo><mi>H</mi><mo></mo><mi>D</mi></math> nên <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>E</mi><mo></mo><mi>H</mi><mo></mo><mi>G</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>90</mn><mo>∘</mo></msup></mstyle></math> (**)<br />Chứng minh tương tự:</p>
<p>Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>D</mi><mo></mo><mi>C</mi><mo></mo><mi>B</mi></mrow><mo>^</mo></mover><mo>+</mo><mover accent="true"><mrow><mi>A</mi><mo></mo><mi>D</mi><mo></mo><mi>C</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>180</mn><mn>0</mn></msup></mstyle></math> (hai góc trong cùng phía bù nhau)<br />Mà <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>E</mi><mo></mo><mi>C</mi><mo></mo><mi>D</mi></mrow><mo>^</mo></mover><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mover accent="true"><mrow><mi>D</mi><mo></mo><mi>C</mi><mo></mo><mi>B</mi></mrow><mo>^</mo></mover></mstyle></math> (do CE là phân giác góc DCB)<br />Nên <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>E</mi><mo></mo><mi>D</mi><mo></mo><mi>C</mi></mrow><mo>^</mo></mover><mo>+</mo><mover accent="true"><mrow><mi>E</mi><mo></mo><mi>C</mi><mo></mo><mi>D</mi></mrow><mo>^</mo></mover><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo>(</mo><mover accent="true"><mrow><mi>A</mi><mo></mo><mi>D</mi><mo></mo><mi>C</mi></mrow><mo>^</mo></mover><mo>+</mo><mover accent="true"><mrow><mi>D</mi><mo></mo><mi>C</mi><mo></mo><mi>B</mi></mrow><mo>^</mo></mover><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⋅</mo><msup><mn>180</mn><mn>0</mn></msup><mo>=</mo><msup><mn>90</mn><mn>0</mn></msup></mstyle></math><br />Lại có:<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>E</mi><mo></mo><mi>D</mi><mo></mo><mi>C</mi></mrow><mo>^</mo></mover><mo>+</mo><mover accent="true"><mrow><mi>E</mi><mo></mo><mi>C</mi><mo></mo><mi>D</mi></mrow><mo>^</mo></mover><mo>+</mo><mover accent="true"><mrow><mi>D</mi><mo></mo><mi>E</mi><mo></mo><mi>C</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>180</mn><mn>0</mn></msup></mstyle></math> (tổng ba góc trong tam giác DEC)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mover accent="true"><mrow><mi>D</mi><mo></mo><mi>E</mi><mo></mo><mi>C</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>180</mn><mn>0</mn></msup><mo>-</mo><mrow><mo>(</mo><mover accent="true"><mrow><mi>E</mi><mo></mo><mi>D</mi><mo></mo><mi>C</mi></mrow><mo>^</mo></mover><mo>+</mo><mover accent="true"><mrow><mi>E</mi><mo></mo><mi>C</mi><mo></mo><mi>D</mi></mrow><mo>^</mo></mover><mo>)</mo></mrow><mo>=</mo><msup><mn>180</mn><mn>0</mn></msup><mo>-</mo><msup><mn>90</mn><mn>0</mn></msup><mo>=</mo><msup><mn>90</mn><mn>0</mn></msup></mstyle></math><br />Hay <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>H</mi><mo></mo><mi>E</mi><mo></mo><mi>F</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>90</mn><mn>0</mn></msup><mrow><mo>(</mo><mo>*</mo><mo>*</mo><mo>*</mo><mo>)</mo></mrow></mstyle></math><br />Từ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mrow><mo>(</mo><mmultiscripts><mo>)</mo><mprescripts></mprescripts><mo>*</mo></mmultiscripts></mrow><mo>,</mo><mrow><mo>(</mo><mmultiscripts><mo>)</mo><mprescripts></mprescripts><mrow><mo>*</mo><mo>*</mo></mrow></mmultiscripts></mrow></mstyle></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>*</mo><mo> </mo><mo>*</mo><mo> </mo><mo>*</mo><mo>)</mo></math> ta thấy tứ giác EFGH có ba góc vuông nên<br />là hình chữ nhật (dấu hiệu nhận biết hình chữ nhật)</p>
<p> </p>
Xem lời giải bài tập khác cùng bài
Hướng dẫn giải Bài 58 (Trang 99 SGK Toán Hình học 8, Tập 1)
Xem lời giải
Hướng dẫn giải Bài 59 (Trang 99 SGK Toán Hình học 8, Tập 1)
Xem lời giải
Hướng dẫn giải Bài 60 (Trang 99 SGK Toán Hình học 8, Tập 1)
Xem lời giải
Hướng dẫn giải Bài 61 (Trang 99 SGK Toán Hình học 8, Tập 1)
Xem lời giải
Hướng dẫn giải Bài 62 (Trang 99 SGK Toán Hình học 8, Tập 1)
Xem lời giải
Hướng dẫn giải Bài 63 (Trang 100 SGK Toán Hình học 8, Tập 1)
Xem lời giải
Hướng dẫn giải Bài 65 (Trang 100 SGK Toán Hình học 8, Tập 1)
Xem lời giải
Hướng dẫn giải Bài 66 (Trang 100 SGK Toán Hình học 8, Tập 1)
Xem lời giải