Bài 6: Đối Xứng Trục
Hướng dẫn giải Bài 36 (Trang 87 SGK Toán Hình học 8, Tập 1)
<p><strong>Đề bài</strong><br />Cho góc xOy có số đo <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>50</mn><mo>∘</mo></msup></math>, điểm A nằm trong góc đó. Vẽ điểm B đối xứng với A qua Ox, vẽ điểm C đối xứng với A qua Oy.<br />a) So sánh các độ dài OB và OC.<br />b) Tính số đo góc BOC.</p>
<p><strong>Lời giải chi tiết </strong></p>
<p><img class="wscnph" style="max-width: 100%;" src="https://static.colearn.vn:8413/v1.0/upload/library/01072022/e528a4a5-1097-469f-a93b-32e3668e8b2c.PNG" /></p>
<p>a) Vì <span id="MathJax-Element-18-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></math>"><span id="MJXc-Node-72" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-73" class="mjx-mrow"><span id="MJXc-Node-74" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span></span></span></span> đối xứng với <span id="MathJax-Element-19-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-75" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-76" class="mjx-mrow"><span id="MJXc-Node-77" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span></span></span></span> qua <span id="MathJax-Element-20-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>O</mi><mi>x</mi></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>x</mi></math></span></span> (giả thiết)</p>
<p><span id="MathJax-Element-21-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"><mo stretchy="false">&#x21D2;</mo><mi>O</mi><mi>x</mi></math>"><span id="MJXc-Node-82" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-83" class="mjx-mrow"><span id="MJXc-Node-84" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">⇒</span></span><span id="MJXc-Node-85" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I">O</span></span><span id="MJXc-Node-86" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">x</span></span></span></span></span> là đường trung trực của <span id="MathJax-Element-22-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></math>"><span id="MJXc-Node-87" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-88" class="mjx-mrow"><span id="MJXc-Node-89" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-90" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span></span></span></span></p>
<p><span id="MathJax-Element-23-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"><mo stretchy="false">&#x21D2;</mo><mi>O</mi><mi>A</mi><mo>=</mo><mi>O</mi><mi>B</mi></math>"><span id="MJXc-Node-91" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-92" class="mjx-mrow"><span id="MJXc-Node-93" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">⇒</span></span><span id="MJXc-Node-94" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I">O</span></span><span id="MJXc-Node-95" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-96" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">=</span></span><span id="MJXc-Node-97" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I">O</span></span><span id="MJXc-Node-98" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span></span></span></span> (tính chất đường trung trực) (1)</p>
<p>Vì <span id="MathJax-Element-24-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></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></span></span> đối xứng với <span id="MathJax-Element-25-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> qua <span id="MathJax-Element-26-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>O</mi><mi>y</mi></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>y</mi></math></span></span> <span id="MathJax-Element-27-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"><mo stretchy="false">&#x21D2;</mo><mi>O</mi><mi>y</mi></math>"><span id="MJXc-Node-109" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-110" class="mjx-mrow"><span id="MJXc-Node-111" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">⇒</span></span><span id="MJXc-Node-112" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I">O</span></span><span id="MJXc-Node-113" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">y</span></span></span></span></span> là đường trung trực của <span id="MathJax-Element-28-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>C</mi></math>"><span id="MJXc-Node-114" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-115" class="mjx-mrow"><span id="MJXc-Node-116" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">A</span></span><span id="MJXc-Node-117" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span></span></span></span></p>
<p><span id="MathJax-Element-29-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"><mo stretchy="false">&#x21D2;</mo><mi>O</mi><mi>A</mi><mo>=</mo><mi>O</mi><mi>C</mi></math>"><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">⇒</mo><mi>O</mi><mi>A</mi><mo>=</mo><mi>O</mi><mi>C</mi></math></span></span> (tính chất đường trung trực) (2)</p>
<p>Từ (1) và (2) suy ra <span id="MathJax-Element-30-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>O</mi><mi>B</mi><mo>=</mo><mi>O</mi><mi>C</mi><mo>.</mo></math>"><span id="MJXc-Node-126" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-127" class="mjx-mrow"><span id="MJXc-Node-128" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">O</span></span><span id="MJXc-Node-129" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">B</span></span><span id="MJXc-Node-130" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">=</span></span><span id="MJXc-Node-131" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I">O</span></span><span id="MJXc-Node-132" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I">C</span></span><span id="MJXc-Node-133" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R">.</span></span></span></span></span></p>
<p>b) Vì OA=OB (chứng minh trên) <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>O</mi><mo>⁢</mo><mi>B</mi></mstyle></math> cân tại O (dấu hiệu nhận biết tam giác cân).</p>
<p>Trong tam giác cân đường trung trực của cạnh đáy đồng thời là đường phân giác nên Ox là phân giác của <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>A</mi><mo>⁢</mo><mi>O</mi><mo>⁢</mo><mi>B</mi></mrow><mo>^</mo></mover><mo>.</mo></math><br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mover accent="true"><msub><mi>O</mi><mn>1</mn></msub><mo>^</mo></mover><mo>=</mo><mover accent="true"><msub><mi>O</mi><mn>2</mn></msub><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>O</mi><mo>⁢</mo><mi>B</mi></mrow><mo>^</mo></mover><mo>⁢</mo></math><br />Lại có <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>A</mi><mo>⁢</mo><mi>O</mi><mo>⁢</mo><mi>C</mi></mstyle></math> cân tại O (vì OA=OC)<br />Trong tam giác cân đường trung trực của cạnh đáy đồng thời là đường phân giác nên Oy là phân giác của <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>A</mi><mo>⁢</mo><mi>O</mi><mo>⁢</mo><mi>C</mi></mrow><mo>^</mo></mover><mo>.</mo></math><br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mover accent="true"><msub><mi>O</mi><mn>3</mn></msub><mo>^</mo></mover><mo>=</mo><mover accent="true"><msub><mi>O</mi><mn>4</mn></msub><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>O</mi><mo>⁢</mo><mi>C</mi></mrow><mo>^</mo></mover><mo>⁢</mo></math><br />Do đó<br /><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⁢</mo><mover accent="true"><mrow><mi>A</mi><mo>⁢</mo><mi>O</mi><mo>⁢</mo><mi>B</mi></mrow><mo>^</mo></mover><mo>+</mo><mover accent="true"><mrow><mi>A</mi><mo>⁢</mo><mi>O</mi><mo>⁢</mo><mi>C</mi></mrow><mo>^</mo></mover><mo>=</mo><mover accent="true"><msub><mi>O</mi><mn>1</mn></msub><mo>^</mo></mover><mo>+</mo><mover accent="true"><msub><mi>O</mi><mn>2</mn></msub><mo>^</mo></mover><mo>+</mo><mover accent="true"><msub><mi>O</mi><mn>3</mn></msub><mo>^</mo></mover><mo>+</mo><mover accent="true"><msub><mi>O</mi><mn>4</mn></msub><mo>^</mo></mover><mo>=</mo><mover accent="true"><msub><mi>O</mi><mn>1</mn></msub><mo>^</mo></mover><mo>+</mo><mover accent="true"><msub><mi>O</mi><mn>1</mn></msub><mo>^</mo></mover><mo>+</mo><mover accent="true"><msub><mi>O</mi><mn>3</mn></msub><mo>^</mo></mover><mo>+</mo><mpadded><mover accent="true"><msub><mi>O</mi><mn>3</mn></msub><mo>^</mo></mover></mpadded><mo>⁢</mo><mspace linebreak="newline"/><mo>=</mo><mn>2</mn><mo>.</mo><mover accent="true"><msub><mi>O</mi><mn>1</mn></msub><mo>^</mo></mover><mo>+</mo><mn>2</mn><mo>.</mo><mover accent="true"><msub><mi>O</mi><mn>3</mn></msub><mo>^</mo></mover><mo>=</mo><mn>2</mn><mo>⁢</mo><mrow><mo>(</mo><mover accent="true"><msub><mi>O</mi><mn>1</mn></msub><mo>^</mo></mover><mo>+</mo><mover accent="true"><msub><mi>O</mi><mn>3</mn></msub><mo>^</mo></mover><mo>)</mo></mrow><mo>=</mo><mn>2</mn><mover accent="true"><mrow><mi>x</mi><mo>⁢</mo><mi>O</mi><mo>⁢</mo><mi>y</mi></mrow><mo>^</mo></mover><mo>⁢</mo><mo>=</mo><mpadded><msup><mn>2.50</mn><mo>∘</mo></msup></mpadded><mo>=</mo><msup><mn>100</mn><mo>∘</mo></msup><mspace linebreak="newline"/><mi>V</mi><mi>ậ</mi><mi>y</mi><mo> </mo><mover accent="true"><mrow><mi>B</mi><mo>⁢</mo><mi>O</mi><mo>⁢</mo><mi>C</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>100</mn><mo>∘</mo></msup><mo>⁢</mo></math></p>
<p><br /><br /></p>
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