Bài 3: Diện Tích Tam Giác
Hướng dẫn giải Bài 24 (Trang 123 SGK Toán Hình học 8, Tập 1)
<p><strong class="content_question">Đề bài</strong></p>
<p>Tính diện tích tam giác cân có cạnh đáy bằng <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></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></span></span> và cạnh bên bằng <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>b</mi><mo>.</mo></math>"><span id="MJXc-Node-4" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5" class="mjx-mrow"><span id="MJXc-Node-6" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">b</span></span><span id="MJXc-Node-7" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">.</span></span></span></span></span></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>b</mi><mo>.</mo></math>"><span class="mjx-math" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">Lời giải chi tiết</span></span></span></span></span></p>
<p>Gọi <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>h</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">h</span></span></span></span></span> là chiều cao của tam giác cân có đáy là <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></math>"><span id="MJXc-Node-11" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-12" class="mjx-mrow"><span id="MJXc-Node-13" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">a</span></span></span></span></span> và cạnh bên là <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>b</mi><mo>.</mo></math>"><span id="MJXc-Node-14" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-15" class="mjx-mrow"><span id="MJXc-Node-16" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">b</span></span><span id="MJXc-Node-17" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">.</span></span></span></span></span> </p>
<p>Xét tam 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>A</mi><mi>B</mi><mi>C</mi></math>"><span id="MJXc-Node-18" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-19" class="mjx-mrow"><span id="MJXc-Node-20" 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><mi>C</mi></math></span></span> cân tại <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>A</mi></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span> có <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><mo>=</mo><mi>b</mi><mo>,</mo><mi>B</mi><mi>C</mi><mo>=</mo><mi>a</mi></math>"><span id="MJXc-Node-26" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-27" class="mjx-mrow"><span id="MJXc-Node-28" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-29" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span><span id="MJXc-Node-30" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">=</span></span><span id="MJXc-Node-31" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I">b</span></span><span id="MJXc-Node-32" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">,</span></span><span id="MJXc-Node-33" class="mjx-mi MJXc-space1"><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-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">=</span></span><span id="MJXc-Node-36" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I">a</span></span></span></span></span> và chiều cao <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>A</mi><mi>H</mi><mo>=</mo><mi>h</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">A</span></span><span id="MJXc-Node-40" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">H</span></span><span id="MJXc-Node-41" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">=</span></span><span id="MJXc-Node-42" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I">h</span></span></span></span></span>. Ta tính diện tích tam giác <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>A</mi><mi>B</mi><mi>C</mi></math>"><span id="MJXc-Node-43" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-44" class="mjx-mrow"><span id="MJXc-Node-45" 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><mi>C</mi></math></span></span>.</p>
<p><img src="https://img.loigiaihay.com/picture/2018/0716/b24-trang-123-sgk-toan-8-t-1-c2.jpg" /></p>
<p>Vi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi mathvariant="normal">△</mi><mo>⁢</mo><munder accentunder="true"><mrow><mi>A</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>C</mi></mrow><mo>¯</mo></munder></mstyle></math> cân tại <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder accentunder="true"><mi>A</mi><mo>¯</mo></munder></mstyle></math> (gt) nên <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder accentunder="true"><mrow><mi>A</mi><mo>⁢</mo><mi>H</mi></mrow><mo>¯</mo></munder></mstyle></math> vừa là đường cao vửa là đường trung tuyến (tính chất tam giác cân). Suy ra, H là trung điểm của BC<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mo>⇒</mo><mi>B</mi><mi>H</mi><mo>∣</mo><mo>=</mo><mfrac><mrow><mi>B</mi><mo>⁢</mo><msup><mi>C</mi><mi>C</mi></msup></mrow><mn>2</mn></mfrac><mo>=</mo><mfrac><mi>a</mi><mn>2</mn></mfrac></mstyle></math><br />Áp dụng định lý Pytago vào tam giác vuông ABH ta có:<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⁢</mo><mi>A</mi><mo>⁢</mo><msup><mi>B</mi><mn>2</mn></msup><mo>=</mo><mi>A</mi><mo>⁢</mo><msup><mi>H</mi><mn>2</mn></msup><mo>+</mo><mi>B</mi><mo>⁢</mo><mpadded><msup><mi>H</mi><mn>2</mn></msup></mpadded><mo>⁢</mo><mspace linebreak="newline"/><mo>⇒</mo><mi>A</mi><mo>⁢</mo><msup><mi>H</mi><mn>2</mn></msup><mo>=</mo><mi>A</mi><mo>⁢</mo><msup><mi>B</mi><mn>2</mn></msup><mo>-</mo><mi>B</mi><mo>⁢</mo><mpadded><msup><mi>H</mi><mn>2</mn></msup></mpadded><mo>⁢</mo><mspace linebreak="newline"/><mo>⁢</mo><msup><mi>h</mi><mn>2</mn></msup><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><msup><mrow><mo>(</mo><mfrac><mi>a</mi><mn>2</mn></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mn>4</mn><mo>⁢</mo><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>⇒</mo><mi>h</mi><mo>=</mo><mfrac><msqrt><mn>4</mn><mo>⁢</mo><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup></msqrt><mn>2</mn></mfrac><mo>⁢</mo></math><br />Diện tích tam giác ABC là:<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⁢</mo><mi>S</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>h</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><mi>a</mi><mo>.</mo><mfrac><msqrt><mn>4</mn><mo>⁢</mo><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup></msqrt><mn>2</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>⁢</mo><mi>a</mi><mo>⁢</mo><msqrt><mn>4</mn><mo>⁢</mo><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup></msqrt><mo>⁢</mo><mo>.</mo></math></p>
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