Bài 4: Diện Tích Hình Thang
Hướng dẫn giải Bài 30 (Trang 126 SGK Toán Hình học 8, Tập 1)
<p><strong class="content_question">Đề bài</strong></p>
<p>Trên hình <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"><mn>143</mn></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-mn"><span class="mjx-char MJXc-TeX-main-R">143</span></span></span></span></span> ta có hình thang <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>B</mi><mi>C</mi><mi>D</mi></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mi>C</mi><mi>D</mi></math></span></span> với đường trung bình <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>E</mi><mi>F</mi></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>F</mi></math></span></span> và hình chữ nhật <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>G</mi><mi>H</mi><mi>I</mi><mi>K</mi><mo>.</mo></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mi>H</mi><mi>I</mi><mi>K</mi><mo>.</mo></math></span></span> Hãy so sánh diện tích hai hình này, từ đó suy ra một cách chứng minh khác về công thức diện tích hình thang. </p>
<p><img src="https://img.loigiaihay.com/picture/2018/0716/b30-trang-126-sgk-toan-8-t-1-c2_1.jpg" /></p>
<p><strong class="content_detail">Lời giải chi tiết</strong></p>
<p><strong class="content_detail"><img src="https://img.loigiaihay.com/picture/2019/0417/h70-bai-30-trang-126-sgk-toan-8-t1.jpg" /></strong></p>
<p><span class="content_detail">Ta có hình thang <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: 400; font-size: 16.94px; letter-spacing: normal; overflow-wrap: normal; word-spacing: 0px; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; color: #000000; font-family: OpenSans, Tahoma, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; 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-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">A</span></span><span id="MJXc-Node-54" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span><span id="MJXc-Node-55" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span><span id="MJXc-Node-56" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">D</span></span></span></span></span> (<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: 400; font-size: 16.94px; letter-spacing: normal; overflow-wrap: normal; word-spacing: 0px; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; color: #000000; font-family: OpenSans, Tahoma, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><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></math>"><span id="MJXc-Node-57" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-58" class="mjx-mrow"><span id="MJXc-Node-59" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-60" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span><span id="MJXc-Node-61" class="mjx-texatom"><span id="MJXc-Node-62" class="mjx-mrow"><span id="MJXc-Node-63" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">/</span></span></span></span><span id="MJXc-Node-64" class="mjx-texatom"><span id="MJXc-Node-65" class="mjx-mrow"><span id="MJXc-Node-66" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">/</span></span></span></span><span id="MJXc-Node-67" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span><span id="MJXc-Node-68" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">D</span></span></span></span></span>), với đường trung bình <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: 400; font-size: 16.94px; letter-spacing: normal; overflow-wrap: normal; word-spacing: 0px; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; color: #000000; font-family: OpenSans, Tahoma, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; 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 class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>F</mi></math></span></span> và hình chữ nhật <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: 400; font-size: 16.94px; letter-spacing: normal; overflow-wrap: normal; word-spacing: 0px; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; color: #000000; font-family: OpenSans, Tahoma, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mi>H</mi><mi>I</mi><mi>K</mi></math>"><span id="MJXc-Node-73" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-74" class="mjx-mrow"><span id="MJXc-Node-75" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">G</span></span><span id="MJXc-Node-76" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">H</span></span><span id="MJXc-Node-77" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">I</span></span><span id="MJXc-Node-78" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">K</span></span></span></span></span></span><span class="content_detail"> như hình vẽ.</span><strong class="content_detail"><br /></strong></p>
<p><span class="content_detail">Xét hai tam giác vuông: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi mathvariant="normal">△</mi><mo>⁢</mo><mi>A</mi><mo>⁢</mo><mi>E</mi><mo>⁢</mo><mi>G</mi></mstyle></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi mathvariant="normal">△</mi><mo>⁢</mo><mi>D</mi><mo>⁢</mo><mi>E</mi><mo>⁢</mo><mi>K</mi></mstyle></math> có:<br />+) AE=ED (do E là trung điểm của AD)<br />+) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>A</mi><mo>⁢</mo><mi>E</mi><mo>⁢</mo><mi>G</mi></mrow><mo>^</mo></mover><mo>=</mo><mover accent="true"><mrow><mi>D</mi><mo>⁢</mo><mi>E</mi><mo>⁢</mo><mi>K</mi></mrow><mo>^</mo></mover></mstyle></math> (đối đỉnh)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>A</mi><mo>⁢</mo><mi>E</mi><mo>⁢</mo><mi>G</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>D</mi><mo>⁢</mo><mi>E</mi><mo>⁢</mo><mi>K</mi></mstyle></math> (cạnh huyền-góc nhọn)<br />Suy ra <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><msub><mi>S</mi><mrow><mi>A</mi><mo>⁢</mo><mi>E</mi><mo>⁢</mo><mi>G</mi></mrow></msub><mo>=</mo><msub><mi>S</mi><mrow><mi>D</mi><mo>⁢</mo><mi>E</mi><mo>⁢</mo><mi>K</mi></mrow></msub></mstyle></math><br />Xét hai tam giác vuông: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi mathvariant="normal">△</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>F</mi><mo>⁢</mo><mi>H</mi></mstyle></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>C</mi><mo>⁢</mo><mi>F</mi><mo>⁢</mo><mi>I</mi></mstyle></math> có:<br />+) BF=FC (do F là trung điểm của BC)<br />+) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mover accent="true"><mrow><mi>B</mi><mo>⁢</mo><mi>F</mi><mo>⁢</mo><mi>H</mi></mrow><mo>^</mo></mover><mo>=</mo><mover accent="true"><mrow><mi>C</mi><mo>⁢</mo><mi>F</mi><mo>⁢</mo><mi>I</mi></mrow><mo>^</mo></mover></mstyle></math> (đối đỉnh)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>F</mi><mo>⁢</mo><mi>H</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>C</mi><mo>⁢</mo><mi>F</mi><mo>⁢</mo><mi>I</mi></mstyle></math> (cạnh huyền-góc nhọn)<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><msub><mi>S</mi><mrow><mi>B</mi><mo>⁢</mo><mi>F</mi><mo>⁢</mo><mi>H</mi></mrow></msub><mo>=</mo><msub><mi>S</mi><mrow><mi>C</mi><mo>⁢</mo><mi>F</mi><mo>⁢</mo><mi>I</mi></mrow></msub></mstyle></math><br />Do đó <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><msub><mi>S</mi><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>C</mi><mo>⁢</mo><mi>D</mi></mrow></msub><mo>=</mo><msub><mi>S</mi><mrow><mi>A</mi><mo>⁢</mo><mi>E</mi><mo>⁢</mo><mi>K</mi><mo>⁢</mo><mi>I</mi><mo>⁢</mo><mi>F</mi><mo>⁢</mo><mi>B</mi></mrow></msub><mo>+</mo><msub><mi>S</mi><mrow><mi>D</mi><mo>⁢</mo><mi>E</mi><mo>⁢</mo><mi>K</mi></mrow></msub><mo>+</mo><msub><mi>S</mi><mrow><mi>C</mi><mo>⁢</mo><mi>F</mi><mo>⁢</mo><mi>I</mi></mrow></msub></mstyle></math><br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>=</mo><msub><mi>S</mi><mrow><mi>A</mi><mo>⁢</mo><mi>E</mi><mo>⁢</mo><mi>K</mi><mo>⁢</mo><mi>I</mi><mo>⁢</mo><mi>F</mi><mo>⁢</mo><mi>B</mi></mrow></msub><mo>+</mo><msub><mi>S</mi><mrow><mi>A</mi><mo>⁢</mo><mi>E</mi><mo>⁢</mo><mi>G</mi></mrow></msub><mo>+</mo><msub><mi>S</mi><mrow><mi>B</mi><mo>⁢</mo><mi>F</mi><mo>⁢</mo><mi>H</mi></mrow></msub><mo>=</mo><msub><mi>S</mi><mrow><mi>G</mi><mo>⁢</mo><mi>H</mi><mo>⁢</mo><mi>I</mi><mo>⁢</mo><mi>K</mi></mrow></msub></mstyle></math><br />Nên:<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><msub><mi>S</mi><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>C</mi><mo>⁢</mo><mi>D</mi></mrow></msub><mo>=</mo><msub><mi>S</mi><mrow><mi>G</mi><mo>⁢</mo><mi>H</mi><mo>⁢</mo><mi>I</mi><mo>⁢</mo><mi>K</mi></mrow></msub><mo>=</mo><mi>G</mi><mo>⁢</mo><mi>H</mi><mo>.</mo><mi>H</mi><mo>⁢</mo><mi>I</mi><mo>=</mo><mi>E</mi><mo>⁢</mo><mi>F</mi><mo>.</mo><mi>H</mi><mo>⁢</mo><mi>I</mi></mstyle></math> (tính chất đường trung bình hình thang $\mathrm{ABCD})$<br />Do đó <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><msub><mi>S</mi><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>C</mi><mo>⁢</mo><mi>D</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi><mo>+</mo><mi>C</mi><mo>⁢</mo><mi>D</mi></mrow><mn>2</mn></mfrac><mo>.</mo><mi>H</mi><mo>⁢</mo><mi>I</mi></mstyle></math><br />Gọi AJ là chiều cao của hình thang ABCD thì AJ=HI, từ đó suy ra:<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><msub><mi>S</mi><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>C</mi><mo>⁢</mo><mi>D</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi><mo>+</mo><mi>C</mi><mo>⁢</mo><mi>D</mi></mrow><mn>2</mn></mfrac><mo>⋅</mo><mi>A</mi><mo>⁢</mo><mi>J</mi></mstyle></math><br /></span></p>
<p><span class="content_detail">Vậy ta gặp lại công thức tính diện tích hình thang đã được học nhưng bằng một phương pháp chứng minh khác. Mặt khác, ta phát hiện công thức mới : Diện tích hình thang bằng tích của đường trung bình hình thang với chiều cao.<br /><br /></span></p>
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