Bài 5: Diện Tích Hình Thoi
Hướng dẫn giải Bài 34 (Trang 128 SGK Toán Hình học 8, Tập 1)
<p><strong class="content_question">Đề bài</strong></p>
<p>Cho một hình chữ nhật. Vẽ tứ giác có các đỉnh là trung điểm các cạnh của hình chữ nhật . Vì sao tứ giác này là một hình thoi? So sánh diện tích hình thoi và diện tích hình chữ nhật, từ đó suy ra cách tính diện tích hình thoi.</p>
<p><strong class="content_detail">Lời giải chi tiết</strong></p>
<p>Vẽ 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>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>. <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>M</mi><mo>,</mo><mi>N</mi><mo>,</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo>.</mo></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">M</span></span><span id="MJXc-Node-42" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-43" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I">N</span></span><span id="MJXc-Node-44" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-45" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I">P</span></span><span id="MJXc-Node-46" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-47" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I">Q</span></span><span id="MJXc-Node-48" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">.</span></span></span></span></span> lần lượt là trung điểm các cạnh AD,AB,BC,CD</p>
<p>Vẽ tứ giác <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><mi>N</mi><mi>P</mi><mi>Q</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">M</span></span><span id="MJXc-Node-52" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">N</span></span><span id="MJXc-Node-53" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">P</span></span><span id="MJXc-Node-54" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">Q</span></span></span></span></span></p>
<p><img src="https://img.loigiaihay.com/picture/2018/0716/b34-trang-128-sgk-toan-8-t-1-c2.jpg" /></p>
<p>Ta có:</p>
<p><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>M</mi><mi>N</mi></math>"><span id="MJXc-Node-55" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-56" class="mjx-mrow"><span id="MJXc-Node-57" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">M</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span> là đường trung bình của tam 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>A</mi><mi>B</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>D</mi></math></span></span> nên</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi>M</mi><mo>⁢</mo><mi>N</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>D</mi></mstyle></math> (tính chất)<br />PQ là đường trung bình của tam giác CBD nên <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi>P</mi><mo>⁢</mo><mi>Q</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>D</mi></mstyle></math> (tính chất)<br />NP là đường trung bình của tam giác ABC nên <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi>N</mi><mo>⁢</mo><mi>P</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><mi>A</mi><mo>⁢</mo><mi>C</mi></mstyle></math> (tính chất)<br />MQ là đường trung bình của tam giác ADC nên <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi>M</mi><mo>⁢</mo><mi>Q</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><mi>A</mi><mo>⁢</mo><mi>C</mi></mstyle></math> (tính chất)<br />Mà <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi>ABCD</mi></mstyle></math> là hình chữ nhật nên AC=BD (tính chất hình chữ' nhật) nên suy ra MN=PQ=NP=MQ<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>P</mi><mo>⁢</mo><mi>Q</mi></mstyle></math> là hình thoi (dấu hiệu nhận biết hình thoi)<br />Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>A</mi><mo>⁢</mo><mi>M</mi><mo>⁢</mo><mi>N</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>I</mi><mo>⁢</mo><mi>N</mi><mo>⁢</mo><mi>M</mi><mo>,</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>P</mi><mo>⁢</mo><mi>N</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>N</mi><mo>⁢</mo><mi>I</mi><mo>⁢</mo><mi>P</mi><mo>,</mo></math><br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>P</mi><mo>⁢</mo><mi>C</mi><mo>⁢</mo><mi>Q</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>I</mi><mo>⁢</mo><mi>Q</mi><mo>⁢</mo><mi>P</mi><mo>,</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>D</mi><mo>⁢</mo><mi>M</mi><mo>⁢</mo><mi>Q</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>I</mi><mo>⁢</mo><mi>Q</mi><mo>⁢</mo><mi>M</mi></mstyle></math><br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mrow><mo>⇒</mo><msub><mi>S</mi><mrow><mi>A</mi><mo>⁢</mo><mi>M</mi><mo>⁢</mo><mi>N</mi></mrow></msub><mo>=</mo><msub><mi>S</mi><mrow><mi>I</mi><mo>⁢</mo><mi>N</mi><mo>⁢</mo><mi>M</mi></mrow></msub></mrow><mo>,</mo><msub><mi>S</mi><mrow><mi>B</mi><mo>⁢</mo><mi>P</mi><mo>⁢</mo><mi>N</mi></mrow></msub><mo>=</mo><msub><mi>S</mi><mrow><mi>N</mi><mo>⁢</mo><mi>I</mi><mo>⁢</mo><mi>P</mi></mrow></msub></mstyle></math><br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><msub><mi>S</mi><mrow><mi>P</mi><mo>⁢</mo><mi>C</mi><mo>⁢</mo><mi>Q</mi></mrow></msub><mo>=</mo><msub><mi>S</mi><mrow><mi>I</mi><mo>⁢</mo><mi>Q</mi><mo>⁢</mo><mi>P</mi></mrow></msub><mo>,</mo><msub><mi>S</mi><mrow><mi>D</mi><mo>⁢</mo><mi>M</mi><mo>⁢</mo><mi>Q</mi></mrow></msub><mo>=</mo><msub><mi>S</mi><mrow><mi>I</mi><mo>⁢</mo><mi>Q</mi><mo>⁢</mo><mi>M</mi></mrow></msub></mstyle></math><br />Ta có:<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><msub><mi>S</mi><mrow><mi>M</mi><mo>⁢</mo><mi>N</mi><mo>⁢</mo><mi>P</mi><mo>⁢</mo><mi>Q</mi></mrow></msub><mo>=</mo><msub><mi>S</mi><mrow><mi>M</mi><mo>⁢</mo><mi>N</mi><mo>⁢</mo><mi>I</mi></mrow></msub><mo>+</mo><msub><mi>S</mi><mrow><mi>N</mi><mo>⁢</mo><mi>I</mi><mo>⁢</mo><mi>P</mi></mrow></msub><mo>+</mo><msub><mi>S</mi><mrow><mi>I</mi><mo>⁢</mo><mi>Q</mi><mo>⁢</mo><mi>P</mi></mrow></msub><mo>+</mo><msub><mi>S</mi><mrow><mi>M</mi><mo>⁢</mo><mi>Q</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>M</mi><mo>⁢</mo><mi>N</mi></mrow></msub><mo>+</mo><msub><mi>S</mi><mrow><mi>B</mi><mo>⁢</mo><mi>N</mi><mo>⁢</mo><mi>P</mi></mrow></msub><mo>+</mo><msub><mi>S</mi><mrow><mi>P</mi><mo>⁢</mo><mi>C</mi><mo>⁢</mo><mi>Q</mi></mrow></msub><mo>+</mo><msub><mi>S</mi><mrow><mi>M</mi><mo>⁢</mo><mi>Q</mi><mo>⁢</mo><mi>D</mi></mrow></msub></mstyle></math><br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><msub><mi>S</mi><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>C</mi><mo>⁢</mo><mi mathvariant="normal">D</mi></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⋅</mo><mi>A</mi><mo>⁢</mo><mi>B</mi><mo>.</mo><mi>A</mi><mo>⁢</mo><mi>D</mi></mstyle></math><br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⋅</mo><mi>M</mi><mo>⁢</mo><mi>P</mi><mo>⋅</mo><mi>N</mi><mo>⁢</mo><mi>Q</mi></mstyle></math><br />Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>⁢</mo><mi>N</mi><mo>⁢</mo><mi>P</mi><mo>⁢</mo><mi>Q</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><mi>M</mi><mo>⁢</mo><mi>P</mi><mo>.</mo><mi>N</mi><mo>⁢</mo><mi>Q</mi></math>.</p>
<p>Do đó diện tích hình thoi bằng nửa tích hai đường chéo.<br /><br /></p>
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