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223 lines
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HTML
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<h1 class="center">Distance Between 2 Points</h1>
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<div class="fun">
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<p style="float:right; margin: 10px;"><img src="images/dist-pythag.svg" alt="distance by pythagoras" height="59" width="109"></p>
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<h3>Quick Explanation</h3>
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<p>When we know the <b>horizontal</b> and <b>vertical</b> distances between two points we can calculate the straight line distance like this:</p>
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<p class="center larger">distance = <span style="font-size:120%; ">√</span><span class="overline"> a<sup>2</sup> + b<sup>2</sup> </span></p>
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</div>
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<div style="clear:both"></div>
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<p> </p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/dist-2-points-a.svg" alt="graph 2 points" height="181" width="248"></p>
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<p>Imagine you know the location of two points (A and B) like here.</p>
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<p>What is the distance between them?</p>
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<div style="clear:both"></div>
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<p> </p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/dist-2-points-b.gif" alt="graph 2 points" height="192" width="225"></p>
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<p>We can run lines down from <span class="large">A</span>, and along from <span class="large">B</span>, to make a <a href="../right_angle_triangle.html">Right Angled Triangle</a>.</p>
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<p>And with a little help from <a href="../pythagoras.html">Pythagoras</a> we know that:</p>
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<p class="center"><span class="large">a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></span></p>
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<div style="clear:both"></div>
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<p> </p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/dist-2-points-c.gif" alt="graph 2 points" height="192" width="252"></p>
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<p>Now label the <a href="../data/cartesian-coordinates.html">coordinates</a> of points A and B.</p>
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<p class="center80" align="center"><span class="large">x<sub>A</sub></span> means the x-coordinate of point <span class="large">A</span><br>
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<span class="large">y<sub>A</sub></span> means the y-coordinate of point <span class="large">A</span></p>
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<p>The horizontal distance <span class="large"><b>a</b></span> is <span class="large"><b>(x<sub>A</sub> − x<sub>B</sub>)</b></span></p>
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<p>The vertical distance <span class="large"><b>b</b></span> is <span class="large"><b>(y<sub>A</sub> − y<sub>B</sub>)</b></span></p>
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<p> </p>
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<p>Now we can solve for <span class="large">c</span> (the distance between the points):</p>
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<div class="simple">
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="large">c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup></span></div>
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<div class="row"><span class="left">Put in the calculations for a and b:</span><span class="large">c<sup>2</sup> = (x<sub>A</sub> − x<sub>B</sub>)<sup>2</sup> + (y<sub>A</sub> − y<sub>B</sub>)<sup>2</sup></span></div>
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<div class="row"><span class="left">Square root of both sides:</span><span class="large">c = <span style="font-size:120%; ">√</span><span class="overline">(x<sub>A</sub> − x<sub>B</sub>)<sup>2</sup> + (y<sub>A</sub> − y<sub>B</sub>)<sup>2</sup></span></span></div>
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</div>
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</div>
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Done! <h2>Examples</h2>
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<h3>Example 1</h3>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/dist-2-points-d.gif" alt="graph 2 points" height="192" width="225"></p>
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<div class="tbl">
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<div class="row">
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<span class="lt">Fill in the values:</span>
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<span class="rt">c = <span style="font-size:120%; ">√</span><span class="overline">(9 − 3)<sup>2</sup> + (7 − 2)<sup>2</sup></span></span>
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</div>
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<div class="row">
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<span class="lt">Calculate:</span>
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<span class="rtlt">c = <span style="font-size:120%; ">√</span><span class="overline">6<sup>2</sup> + 5<sup>2</sup></span><br>
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c = <span style="font-size:120%; ">√</span><span class="overline">36 + 25</span><br>
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c = <span style="font-size:120%; ">√</span><span class="overline">61</span><br>
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c = 7.8102...</span>
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</div>
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</div>
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<div style="clear:both"></div>
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<h3>Example 2</h3>
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<p>It doesn't matter what order the points are in, because squaring removes any negatives:</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/dist-2-points-e.gif" alt="graph 2 points" height="191" width="225"></p>
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<div class="tbl">
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<div class="row">
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<span class="lt">Fill in the values:</span>
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<span class="rt">c = <span style="font-size:120%; ">√</span><span class="overline">(3 − 9)<sup>2</sup> + (2 − 7)<sup>2</sup></span></span>
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</div>
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<div class="row">
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<span class="lt">Calculate:</span>
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<span class="rtlt">c = <span style="font-size:120%; ">√</span><span class="overline">(−6)<sup>2</sup> + (−5)<sup>2</sup></span><br>
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c = <span style="font-size:120%; ">√</span><span class="overline">36 + 25</span><br>
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c = <span style="font-size:120%; ">√</span><span class="overline">61</span><br>
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c = 7.8102...</span>
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</div>
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</div>
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<div style="clear:both"></div>
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<h3>Example 3</h3>
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<p>And here is another example with some negative coordinates ... it all still works:</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/dist-2-points-f.gif" alt="graph 2 points" height="208" width="262"></p>
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<p> </p>
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<div class="tbl">
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<div class="row">
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<span class="lt">Fill in the values:</span>
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<span class="rt">c = <span style="font-size:120%; ">√</span><span class="overline">(−3 − 7)<sup>2</sup> + (5 − (−1))<sup>2</sup></span></span>
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</div>
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<div class="row">
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<span class="lt">Calculate:</span>
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<span class="rtlt">c = <span style="font-size:120%; ">√</span><span class="overline">(−10)<sup>2</sup> + 6<sup>2</sup></span><br>
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c = <span style="font-size:120%; ">√</span><span class="overline">100 + 36</span><br>
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c = <span style="font-size:120%; ">√</span><span class="overline">136</span><br>
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c = 11.66...</span>
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</div>
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</div>
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<p><i>(Note √136 can be further simplified to 2√34 if you want)</i></p>
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<div style="clear:both"></div>
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<h2>Try It Yourself</h2>
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<p>Drag the points:</p>
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<div class="script" style="height: 360px;">
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images/dist2pts.js
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</div>
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<h2>Three or More Dimensions</h2>
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<p>It works perfectly well in 3 (or more!) dimensions.</p>
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<p>Square the difference for <b>each axis</b>, then sum them up and take the square root:</p>
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<p class="center larger">Distance = <span style="font-size:120%; ">√</span><span class="overline">(x<sub>A</sub> − x<sub>B</sub>)<sup>2</sup> + (y<sub>A</sub> − y<sub>B</sub>)<sup>2</sup> + (z<sub>A</sub> − z<sub>B</sub>)<sup>2</sup></span></p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/dist-2-points-3d.svg" alt="distance between (9,2,7) and (4,8,10) in 3d" height="348" width="401"></p>
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<h3>Example: the distance between the two points (8,2,6) and (3,5,7) is:</h3>
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<p class="large">
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= <span style="font-size:120%; ">√</span><span class="overline">(8−3)<sup>2</sup> + (2−5)<sup>2</sup> + (6−7)<sup>2</sup></span><br>
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= <span style="font-size:120%; ">√</span><span class="overline">5<sup>2</sup> + (−3)<sup>2</sup> + (−1)<sup>2</sup></span><br>
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= <span style="font-size:120%; ">√</span><span class="overline">25 + 9 + 1</span><br>
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= <span style="font-size:120%; ">√</span><span class="overline">35</span></p>
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<p>Which is about <b>5.9</b></p>
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<p>Read more at <a href="../geometry/pythagoras-3d.html">Pythagoras' Theorem in 3D</a><br> </p>
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<div style="clear:both"></div>
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<p>
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</p>
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<div class="questions">513, 514, 1148, 1149, 2994, 2995, 2996, 2997, 4034, 4035</div>
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<div class="related">
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<a href="line-equation-2points.html">Equation of a Line from 2 Points</a>
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<a href="index.html">Algebra Index</a>
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