Bài 7: Hình Bình Hành
Hướng dẫn giải Bài 49 (Trang 93 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><mo>.</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-mo"><span class="mjx-char MJXc-TeX-main-R">.</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>I</mi><mo>,</mo><mi>K</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">I</span></span><span id="MJXc-Node-11" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>K</mi></math></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>C</mi><mi>D</mi><mo>,</mo><mi>A</mi><mi>B</mi><mo>.</mo></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">C</span></span><span id="MJXc-Node-16" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">D</span></span><span id="MJXc-Node-17" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-18" class="mjx-mi MJXc-space1"><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></span></span> 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>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 <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>I</mi><mo>,</mo><mi>C</mi><mi>K</mi></math>"><span id="MJXc-Node-25" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-26" class="mjx-mrow"><span id="MJXc-Node-27" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-28" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">I</span></span><span id="MJXc-Node-29" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-30" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I">C</span></span><span id="MJXc-Node-31" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">K</span></span></span></span></span> theo thứ tự ở <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>M</mi></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</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>N</mi><mo>.</mo></math>"><span id="MJXc-Node-35" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-36" class="mjx-mrow"><span id="MJXc-Node-37" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">N</span></span><span id="MJXc-Node-38" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">.</span></span></span></span></span> Chứng minh rằng:</p>
<p>a) <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>A</mi><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>C</mi><mi>K</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">I</span></span><span id="MJXc-Node-43" class="mjx-texatom"><span id="MJXc-Node-44" class="mjx-mrow"><span id="MJXc-Node-45" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">/</span></span></span></span><span id="MJXc-Node-46" class="mjx-texatom"><span id="MJXc-Node-47" class="mjx-mrow"><span id="MJXc-Node-48" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">/</span></span></span></span><span id="MJXc-Node-49" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span><span id="MJXc-Node-50" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">K</span></span></span></span></span></p>
<p>b) <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>D</mi><mi>M</mi><mo>=</mo><mi>M</mi><mi>N</mi><mo>=</mo><mi>N</mi><mi>B</mi></math>"><span id="MJXc-Node-51" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-52" class="mjx-mrow"><span id="MJXc-Node-53" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">D</span></span><span id="MJXc-Node-54" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">M</span></span><span id="MJXc-Node-55" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">=</span></span><span id="MJXc-Node-56" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I">M</span></span><span id="MJXc-Node-57" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">N</span></span><span id="MJXc-Node-58" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">=</span></span><span id="MJXc-Node-59" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I">N</span></span><span id="MJXc-Node-60" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span></span></span></span></p>
<p><strong><span 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-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>M</mi><mo>=</mo><mi>M</mi><mi>N</mi><mo>=</mo><mi>N</mi><mi>B</mi></math>"><span class="mjx-math" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">Lời giải chi tiết</span></span></span></span></span></strong></p>
<p><span 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>M</mi><mo>=</mo><mi>M</mi><mi>N</mi><mo>=</mo><mi>N</mi><mi>B</mi></math>"><span class="mjx-math" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I"><img class="wscnph" style="max-width: 100%;" src="https://static.colearn.vn:8413/v1.0/upload/library/03072022/d34d2227-b12b-4638-be13-018c104de413.PNG" /></span></span></span></span></span></p>
<p>a) Vì ABCD là hình bình hành (giả thiết)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>A</mi><mi>B</mi><mo>=</mo><mi>C</mi><mpadded><mi mathvariant="normal">D</mi></mpadded></mtd></mtr><mtr><mtd><mi>A</mi><mi>B</mi><mo>/</mo><mo>/</mo><mi>C</mi><mi mathvariant="normal">D</mi></mtd></mtr></mtable></mfenced></mstyle></math> (tính chất hình bình hành)<br />Mà I, K theo thứ tự là trung điểm của CD, AB (giả thiết) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mi>A</mi><mo>⁢</mo><mi>K</mi><mo>=</mo><mi>I</mi><mo>⁢</mo><mi>C</mi></mstyle></math><br />Lại có: AB//DC (chứng minh trên) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mi>A</mi><mi>K</mi><mo>/</mo><mo>/</mo><mi>I</mi><mi>C</mi></mstyle></math><br />Tứ giác AICK có:<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>A</mi><mi>K</mi><mo>/</mo><mo>/</mo><mi>I</mi><mpadded><mi>C</mi></mpadded></mtd></mtr><mtr><mtd><mi>A</mi><mi>K</mi><mo>=</mo><mi>I</mi><mi>C</mi></mtd></mtr></mtable></mfenced></math>(chứng minh trên)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo></mstyle></math> Tứ giác AICK là hình bình hành (dấu hiệu nhận biết hình bình hành)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mi>A</mi><mi>I</mi><mo>/</mo><mo>/</mo><mi>C</mi><mi>K</mi></mstyle></math> (tính chất hình bình hành)<br />b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>D</mi><mo>⁢</mo><mi>C</mi><mo>⁢</mo><mi>N</mi></mstyle></math> có DI=IC (chứng minh trên), IM//CN (vì AI//KC)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mi>D</mi><mo>⁢</mo><mi>M</mi><mo>=</mo><mi>M</mi><mo>⁢</mo><mi>N</mi></mstyle></math> (1) (Đường thẳng đi qua trung điểm một cạnh của tam giác và song song với cạnh thứ hai thì đi qua trung điểm của cạnh thứ ba)</p>
<p>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>B</mi><mo>⁢</mo><mi>M</mi></mstyle></math> có AK=KB (chứng minh trên) và KN//AM (vì AI//CK)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mi>M</mi><mo>⁢</mo><mi>N</mi><mo>=</mo><mi>N</mi><mo>⁢</mo><mi>B</mi></mstyle></math>. (2) (Đường thẳng đi qua trung điểm một cạnh của tam giác và song song với cạnh thứ hai thì đi qua trung điểm của cạnh thứ ba)<br />Từ (1) và (2)<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>D</mi><mo>⁢</mo><mi>M</mi><mo>=</mo><mi>M</mi><mo>⁢</mo><mi>N</mi><mo>=</mo><mi>N</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo></math></p>
Xem lời giải bài tập khác cùng bài
Hướng dẫn giải Bài 43 (Trang 92 SGK Toán Hình học 8, Tập 1)
Xem lời giải
Hướng dẫn giải Bài 44 (Trang 92 SGK Toán Hình học 8, Tập 1)
Xem lời giải
Hướng dẫn giải Bài 45 (Trang 92 SGK Toán Hình học 8, Tập 1)
Xem lời giải
Hướng dẫn giải Bài 46 (Trang 92 SGK Toán Hình học 8, Tập 1)
Xem lời giải
Hướng dẫn giải Bài 47 (Trang 93 SGK Toán Hình học 8, Tập 1)
Xem lời giải
Hướng dẫn giải Bài 48 (Trang 93 SGK Toán Hình học 8, Tập 1)
Xem lời giải