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<title>Finding Parallel and Perpendicular Lines</title>
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<h1 align="center">Parallel and Perpendicular Lines</h1>
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<p align="center">How to use <a href="index.html">Algebra</a> to find <a href="../perpendicular-parallel.html">parallel and perpendicular lines</a>.</p>
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<h2>Parallel Lines</h2>
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<p>How do we know when two lines are <b>parallel</b>?</p>
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<div class="def">
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<p class="center large">Their slopes are the same!</p>
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</div>
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<table border="0" align="center">
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<tr>
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<td class="larger"><p>The <a href="../geometry/slope.html">slope</a> is the value <b>m</b> in the <a href="../equation_of_line.html">equation of a line</a>:</p>
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<p align="center"><span class="largest">y = mx + b</span></p></td>
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<td class="larger"> </td>
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<td><img src="images/y-mxpb-graph.svg" width="120" alt="Slope-Intercept Form" /></td>
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</tr>
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</table> <p> </p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/graph-parallel-ex.gif" alt="graph" width="227" height="192" />
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</p>
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<h3>Example: </h3>
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<p>Find the equation of the line that is:</p>
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<ul>
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<li>parallel to <b>y = 2x + 1 </b></li>
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<li>and passes though the point (5,4)</li>
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</ul>
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<p> </p>
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<p>The slope of <b>y=2x+1 </b>is: <span class="large">2</span></p>
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<p><b>The parallel line needs to have the same slope of 2.</b></p>
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<p> </p>
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<p>We can solve it using the <a href="line-equation-point-slope.html">"point-slope" equation of a line</a>:</p>
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<p align="center"><span class="larger">y − y<sub>1</sub> = 2(x − x<sub>1</sub>)</span></p>
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<p>And then put in the point (5,4):</p>
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<p align="center"><span class="large">y − 4 = 2(x − 5)</span></p>
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<p> </p>
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<p>And that answer is OK, but let's also put it in <a href="../equation_of_line.html">y = mx + b</a> form:</p>
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<p align="center"><span class="larger">y − 4 = 2x − 10</span></p>
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<p align="center"><span class="large">y = 2x − 6</span></p>
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</div>
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<h3>Vertical Lines</h3>
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<p>But this does not work for vertical lines ... I explain why at the end.</p>
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<h3>Not The Same Line</h3>
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<p>Be careful! They may be the <b>same line</b> (but with a different equation), and so are <b>not parallel</b>.</p>
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<p>How do we know if they are really the same line? <b>Check their y-intercepts</b> (where they cross the y-axis) as well as their slope:</p>
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<div class="example">
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<h3>Example: is y = 3x + 2 parallel to y − 2 = 3x ?</h3>
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<p>For<b> y = 3x + 2</b>: the slope is 3, and y-intercept is 2</p>
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<p>For <b>y − 2 = 3x</b>: the slope is 3, and y-intercept is 2</p>
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<p>In fact they are the same line and so are not parallel</p>
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</div>
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<h2>Perpendicular Lines</h2>
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<p>Two lines are Perpendicular when they meet at a right angle (90°).</p>
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<p>To find a <span class="center">perpendicular slope:</span></p>
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<div class="def">
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<p class="center">When one line has a slope of <span class="large">m</span>, a perpendicular line has a slope of <span class="intbl large"><em>−1</em><strong>m</strong></span></p>
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</div>
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<p>In other words the <b> negative <a href="../reciprocal.html">reciprocal</a></b> </p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/graph-perp-ex2.gif" alt="graph" width="223" height="190" /></p>
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<h3>Example: </h3>
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<p>Find the equation of the line that is</p>
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<ul>
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<li>perpendicular to <b>y = −4x + 10 </b></li>
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<li>and passes though the point <b>(7,2)</b></li>
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</ul>
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<p> </p>
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<p>The slope of <b>y=−4x+10 </b>is: <span class="large">−4</span></p>
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<p>The <b>negative reciprocal</b> of that slope is:</p>
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<p class="center large">m = <span class="intbl"><em>−1</em><strong>−4</strong></span> = <span class="intbl"><em>1</em><strong>4</strong></span></p>
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<p>So the perpendicular line will have a slope of 1/4:</p>
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<p align="center"><span class="larger">y − y<sub>1</sub> = (1/4)(x − x<sub>1</sub>)</span></p>
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<p>And now put in the point (7,2):</p>
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<p align="center"><span class="large">y − 2 = (1/4)(x − 7)</span></p>
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<p> </p>
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<p>And that answer is OK, but let's also put it in "y=mx+b" form:</p>
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<p align="center"><span class="larger">y − 2 = x/4 − 7/4</span></p>
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<p align="center"><span class="large">y = x/4 + 1/4</span></p>
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</div>
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<h2>Quick Check of Perpendicular</h2>
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<p>When we multiply a slope <span class="large">m</span> by its perpendicular slope <span class="intbl large"><em>−1</em><strong>m</strong></span> we get simply <span class="large">−1</span>.</p>
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<p>So to quickly check if two lines are perpendicular:</p>
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<div class="center80">
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<p align="center" class="large">When we multiply their slopes, we get −1</p>
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</div>
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<p>Like this:</p>
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<div class="example">
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/graph-perp-ex.gif" alt="graph vertical line" width="228" height="193" /></p>
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<p>Are these two lines perpendicular?</p>
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<div class="simple">
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<table width="250" border="0" align="center">
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<tr>
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<td align="center">Line</td>
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<td align="center">Slope</td>
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</tr>
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<tr>
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<td align="center" class="larger">y = 2x + 1</td>
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<td align="center"><b class="larger">2</b></td>
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</tr>
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<tr>
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<td align="center" class="larger">y = −0.5x + 4</td>
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<td align="center"><b class="larger">−0.5</b></td>
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</tr>
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</table>
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</div>
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<p>When we multiply the two slopes we get:</p>
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<p align="center" class="large">2 × (−0.5) = −1 </p>
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<p>Yes, we got −1, so they are perpendicular.</p>
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</div>
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<h2>Vertical Lines</h2>
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<p>The previous methods work nicely except for a <b>vertical line</b>:</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/graph-2-points-vertical.gif" alt="graph vertical line" width="148" height="189" /></p>
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<p> </p>
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<p>In this case the gradient is <b>undefined</b> (as we <a href="../numbers/dividing-by-zero.html">cannot divide by 0</a>):</p>
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<p class="center large">m = <span class="intbl"><em>y<sub>A</sub> − y<sub>B</sub></em><strong>x<sub>A</sub> − x<sub>B</sub></strong></span> = <span class="intbl"><em>4 − 1</em><strong>2 − 2</strong></span> = <span class="intbl"><em>3</em><strong>0</strong></span> = undefined</p>
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<div style="clear:both"></div>
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<p>So just rely on the fact that:</p>
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<ul>
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<li>a vertical line is parallel to another vertical line.</li>
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<li>a vertical line is perpendicular to a horizontal line (and vice versa).</li>
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</ul>
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<h2>Summary</h2>
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<ul>
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<li>parallel lines: <b>same</b> slope</li>
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<li>perpendicular lines: <b>negative reciprocal</b> slope (−1/m)</li>
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</ul>
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<p> </p>
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<script type="text/javascript">getQ(7315, 7355, 7365, 7374, 7375, 7382, 7324, 7364, 7379, 7384);</script>
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