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298 lines
14 KiB
HTML
298 lines
14 KiB
HTML
<!DOCTYPE html>
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<title>Pythagoras Theorem</title>
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<!-- #BeginEditable "Body" -->
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<center>
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<h1>Pythagoras' Theorem</h1>
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</center>
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<p style="float:left; margin: 10px;"><img src="images/pythagoras.jpg" alt="pythagoras" height="118" width="58"><br>
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<span class="tiny"><i>Pythagoras</i></span></p>
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<p class="larger"> </p>
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<p>Over 2000 years ago there was an amazing discovery about triangles:</p>
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<p class="large center"><i> When a triangle has a right angle (90°) ...</i></p>
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<p class="large center"><i>... and squares are made on each
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of the three sides, ...</i></p>
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<div class="script" style="height: 330px;">
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geometry/images/pyth1.js
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</div>
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<p><i>... then the biggest square has the <b>exact same area</b> as the other two squares put together!</i></p><br>
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<div class="clear"></div>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="geometry/images/pythagoras-abc.svg" alt="Pythagoras" height="242" width="221"></p>
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<p>It is called "Pythagoras' Theorem" and can be written in one short equation:</p>
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<p class="center larger">a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></p>
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<p class="center"><img src="geometry/images/pythagoras-squares.svg" alt="pythagoras squares a^2 + b^2 = c^2" height="109" width="375"></p>
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<p>Note:</p>
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<ul>
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<li><b>c</b> is the <span class="large"> <b>longest side</b> of the triangle</span></li>
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<li><b>a</b> and <b>b</b> are the other two sides</li>
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</ul>
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<h2>Definition</h2>
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<p>The longest side of the triangle is called the "hypotenuse", so the formal definition is:</p>
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<div class="def">
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<p class="center">In a right angled triangle:<br>
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the square of the
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hypotenuse is equal to<br>
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the sum of the squares of the other two sides.</p>
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</div>
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<h2>Sure ... ?</h2>
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<p>Let's see if it really works using an example.</p>
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<div class="example">
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<h3>Example: A <a href="geometry/triangle-3-4-5.html">"3, 4, 5" triangle</a> has a right angle in it.</h3>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="geometry/images/triangle-3-4-5-pyth.svg" alt="triangle 3 4 5" height="229" width="209"></td>
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<td><br>
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<p>Let's check if the areas <b>are</b> the same:</p>
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<p class="center larger">3<sup>2</sup> + 4<sup>2</sup> = 5<sup>2</sup></p>
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<p>Calculating this becomes:</p>
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<p class="center larger">9 + 16 = 25</p>
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<p><i>It works ... like Magic!</i></p></td>
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</tr>
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</tbody></table>
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</div>
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<p class="center"><img src="geometry/images/triangle-3-4-5-leg.jpg" alt="triangle 3 4 5 lego" height="216" width="200"></p>
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<h2>Why Is This Useful?</h2>
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<p><span class="large">If we know the lengths of <b>two sides</b> of a right angled triangle, we can find the length of the <b>third side</b>. (But remember it only works on right angled
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triangles!)</span></p>
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<h2>How Do I Use it?</h2>
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<p>Write it down as an equation:</p>
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<div class="clear">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="geometry/images/triangle-abc.svg" alt="abc triangle" height="109" width="189"></td>
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<td valign="middle"> </td>
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<td valign="middle"><span class="large">a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></span></td>
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</tr>
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</tbody></table>
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</div>
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<div class="center"> </div>
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<p><br>
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Then we use <a href="algebra/index.html">algebra</a> to find any missing value, as in these examples:</p>
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<div class="example">
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<h3>Example: Solve this triangle</h3>
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<p class="center"><img src="geometry/images/triangle-5-12-c.svg" alt="right angled triangle 5 12 c" height="87" width="176"></p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></span></div>
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<div class="row"><span class="left">Put in what we know:</span><span class="right">5<sup>2</sup> + 12<sup>2</sup> = c<sup>2</sup></span></div>
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<div class="row"><span class="left">Calculate squares:</span><span class="right">25 + 144 = c<sup>2</sup></span></div>
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<div class="row"><span class="left">25+144=169:</span><span class="right">169 = c<sup>2</sup></span></div>
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<div class="row"><span class="left">Swap sides:</span><span class="right">c<sup>2</sup> = 169</span></div>
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<div class="row"><span class="left">Square root of both sides:</span><span class="right">c = √169 </span></div>
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<div class="row"><span class="left">Calculate:</span><span class="right"><span class="large"><b>c = 13</b> </span></span></div>
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</div>
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</div>
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<p style="float:right; margin: 0 0 5px 30px;"><img src="geometry/images/triangle-build-diag1.svg" alt="Multiples of 3,4,5" height="141" width="307"></p>
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<p>Read <a href="builders-math.html">Builder's Mathematics</a> to see practical uses for this.</p>
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<p>Also read about <a href="square-root.html">Squares and Square Roots</a> to find out why <span class="larger">√</span>169 = 13</p>
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<div class="example">
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<h3>Example: Solve this triangle.</h3>
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<p class="center"><img src="geometry/images/triangle-9-b-15.svg" alt="right angled triangle 9 b 15" height="114" width="142"></p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></span></div>
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<div class="row"><span class="left">Put in what we know:</span><span class="right">9<sup>2</sup> + b<sup>2</sup> = 15<sup>2</sup></span></div>
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<div class="row"><span class="left">Calculate squares:</span><span class="right">81 + b<sup>2</sup> = 225 </span></div>
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<div class="row"><span class="left">Take 81 from both sides: </span><span class="right">81 − 81 + b<sup>2</sup> = 225 − 81</span></div>
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<div class="row"><span class="left">Calculate: </span><span class="right">b<sup>2</sup> = 144</span></div>
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<div class="row"><span class="left">Square root of both sides:</span><span class="right">b = √144 </span></div>
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<div class="row"><span class="left">Calculate:</span><span class="right"><b>b = 12</b> </span></div>
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</div>
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</div>
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<div class="example">
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<h3>Example: What is the diagonal distance across a square of size 1?</h3>
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<p class="center"><img src="images/unit-square-diagonal.gif" alt="Unit Square Diagonal" height="141" width="128"></p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></span></div>
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<div class="row"><span class="left">Put in what we know:</span><span class="right">1<sup>2</sup> + 1<sup>2</sup> = c<sup>2</sup></span></div>
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<div class="row"><span class="left">Calculate squares:</span><span class="right">1 + 1 = c<sup>2</sup></span></div>
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<div class="row"><span class="left">1+1=2: </span><span class="right">2 = c<sup>2</sup></span></div>
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<div class="row"><span class="left">Swap sides: </span><span class="right">c<sup>2</sup> = 2</span></div>
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<div class="row"><span class="left">Square root of both sides:</span><span class="right"><b>c = √2</b></span></div>
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<div class="row"><span class="left">Which is about:</span><span class="right"><b>c = 1.4142...</b></span></div>
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</div>
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</div><br>
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<p>It works the other way around, too: when the three sides of a triangle make <span class="large nobr">a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></span>, then the triangle is right angled.</p>
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<div class="example">
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<h3>Example: Does this triangle have a Right Angle?</h3>
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<p class="center"><img src="geometry/images/triangle-10-24-26.svg" alt="10 24 26 triangle" height="157" width="160"></p>
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<p>Does<span class="large"> a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup> ?</span></p>
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<ul>
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<li>a<sup>2</sup> + b<sup>2</sup> = 10<sup>2</sup> + 24<sup>2</sup> = 100 + 576 =<b> 676</b></li>
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<li>c<sup>2</sup> = 26<sup>2</sup> = <b>676</b></li>
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</ul>
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<p>They are equal, so ...</p>
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<p class="center larger">Yes, it does have a Right Angle!</p>
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</div>
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<div class="example">
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<h3>Example: Does an 8, 15, 16 triangle have a Right Angle?</h3>
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<p class="large"><b>Does 8</b><sup>2</sup> + <b>15</b><sup>2</sup> = <b>16</b><sup>2 </sup>?</p>
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<ul>
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<li>8<sup>2</sup> + 15<sup>2</sup> = 64 + 225 = <b>289</b>,</li>
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<li>but 16<sup>2 </sup>= <b>256</b></li>
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</ul>
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<p class="center larger">So, NO, it does not have a Right Angle</p>
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</div>
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<div class="example">
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<h3>Example: Does this triangle have a Right Angle?</h3>
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<p class="center"><img src="geometry/images/triangle-r3-r5-r8.svg" alt="Triangle with roots" height="151" width="103"></p>
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<p>Does<span class="large"> a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup> ?</span></p>
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<div class="so"><span class="large">Does (<b>√</b>3)<sup>2</sup> + (<b>√</b>5)<sup>2</sup> = (<b>√</b>8)<sup>2</sup> ?</span></div>
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<div class="so"><span class="large">Does 3 + 5 = 8 ?</span></div>
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<p class="center larger">Yes, it does!</p>
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<p>So this <b>is</b> a right-angled triangle</p>
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</div>
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<div class="Clear""></div>
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<h2>And You Can Prove The Theorem Yourself !</h2>
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<p>Get paper pen and scissors, then using the following animation as a guide:</p>
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<p class="center"><iframe src="https://www.youtube.com/embed/_87RbSoELW8?rel=0&showinfo=0" allowfullscreen="" title="Pythagoras proof" height="203" frameborder="0" width="360"></iframe></p>
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<ul>
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<li>Draw a right angled triangle on the paper, leaving plenty of space.</li>
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<li>Draw a square along the hypotenuse (the longest side)</li>
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<li>Draw the same sized square on the other side of the hypotenuse</li>
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<li>Draw lines as shown on the animation, like this:</li>
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<li><img src="images/pythagoras-cutout.png" alt="cut sqaure" height="101" width="102"></li>
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<li>Cut out the shapes</li>
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<li>Arrange them so that you can prove that the big square has the same area as the two squares on the other sides</li>
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</ul>
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<h2>Another, Amazingly Simple, Proof</h2>
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<p>Here is one of the oldest proofs that the square on the long side has the same area as the other squares.</p>
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<p class="center"><iframe src="https://www.youtube.com/embed/Zb0thZ6_5G8?rel=0&showinfo=0" allowfullscreen="" title="Pythagoras proof" height="203" frameborder="0" width="360"></iframe></p>
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<p>Watch the animation, and pay attention when the triangles start sliding around.</p>
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<p>You may want to watch the animation a few times to understand what is happening.</p>
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<p>The purple triangle is the important one.</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="geometry/images/pythagoras-proof-2a.svg" alt="before" height="153" width="153"></td>
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<td> becomes </td>
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<td><img src="geometry/images/pythagoras-proof-2b.svg" alt="before" height="153" width="153"></td>
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</tr>
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</tbody></table>
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<p> </p>
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<p>We also have a <a href="geometry/pythagorean-theorem-proof.html">proof by adding up the areas</a>.</p>
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<div class="history">
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Historical Note: while we call it Pythagoras' Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived.
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</div>
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<p> </p>
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<div class="questions">511,512,617,618, 1145, 1146, 1147, 2359, 2360, 2361</div>
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<div class="activity"> <a href="activity/pythagoras-theorem-shoes.html">Activity: Pythagoras' Theorem</a><br>
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<a href="activity/walk-in-desert.html">Activity: A Walk in the Desert</a> </div>
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<div class="related">
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<a href="right_angle_triangle.html">Right Angled Triangles</a>
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<a href="activity/fishing-rod.html">The Fishing Rod</a>
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<a href="geometry/pythagoras-3d.html">Pythagoras in 3D</a>
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<a href="geometry/pythagoras-general.html">Pythagoras Generalizations</a>
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<a href="triangle.html">Triangles</a>
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<a href="pythagorean_triples.html">Pythagorean Triples</a>
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<a href="geometry/pythagorean-theorem-proof.html">Pythagorean Theorem Algebra Proof</a>
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