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198 lines
7.4 KiB
HTML
198 lines
7.4 KiB
HTML
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<div id="hdr"></div>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">3, 4, 5 Triangle</h1>
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<h2>Need a Right Angle (90°) Fast ... ?</h2>
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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>
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<p class="large">Make a 3,4,5 Triangle !</p>
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<p>Connect three lines:</p>
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<ul>
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<li>3 long</li>
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<li>4 long</li>
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<li>5 long</li>
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</ul>
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<p>And you will have a right angle (90°)</p></td>
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<td><img src="images/triangle-3-4-5.svg" alt="3,4,5 Triangle" height="190" width="226"></td>
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</tr>
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</tbody></table>
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<h2>Other Lengths</h2>
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<p>You can use other lengths by multiplying each side by 2. Or by 10. Or any multiple.</p>
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<p class="center"><img src="images/triangle-3-4-5-n.svg" alt="Multiples of 3,4,5" height="" width=""></p>
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<h2>Drawing It</h2>
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<p>Let us say you need to mark a right angle coming from a point on a wall.</p>
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<p>You decide to use 300, 400 and 500 cm lines.</p>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td><img src="images/triangle-3-4-5-a.gif" alt="triangle base 300" height="150" width="100"></td>
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<td><img src="images/triangle-3-4-5-b.gif" alt="triangle draw 400 arc" height="150" width="132"></td>
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<td><img src="images/triangle-3-4-5-c.gif" alt="triangle draw 500 arc" height="150" width="129"></td>
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<td><img src="images/triangle-3-4-5-d.gif" alt="triangle intersect 400 and 500 arc" height="150" width="128"></td>
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<td><img src="images/triangle-3-4-5-e.gif" alt="triangle 300 400 500 right angle " height="149" width="120"></td>
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</tr>
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<tr>
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<td>Draw a 300 line along the wall</td>
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<td>Draw an arc 400 away from the start of the 300 line</td>
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<td>Draw an arc 500 away from the end of the 300 line</td>
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<td>Connect from the start of the 300 line to where the arcs cross</td>
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<td>And you have your "3,4,5" triangle with its right angle</td>
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</tr>
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</tbody></table>
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<h2>The Mathematics Behind It</h2>
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<p>The <a href="../pythagoras.html">Pythagoras Theorem</a> says:</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="images/triangle-abc.svg" alt="triangle abc" height="109" width="189"></td>
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<td valign="middle">
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<p>In a right-angled triangle, the square of a (a<sup>2</sup>) plus the square of b (b<sup>2</sup>) is equal to the square of c (c<sup>2</sup>):</p>
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<p class="center large">a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></p></td>
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</tr>
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</tbody></table>
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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="images/triangle-3-4-5-pyth.svg" alt="triangle 3,4,5 with areas 9,15,25" height="229" width="209"></td>
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<td><br>
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Let's check if it <b>does</b> work:<br>
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<p class="center large">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 large">9 + 16 = 25</p>
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<p><i>Yes, it works !</i></p></td>
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</tr>
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</tbody></table>
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<p class="center"><img src="images/triangle-3-4-5-leg.jpg" alt="triangle 3,4,5 with areas 9,15,25 lego" height="216" width="200"></p>
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<h2>Other Combinations</h2>
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<p>Yes, there are other combinations that work (not just by multiplying). Here are two others:</p>
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<table align="center">
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<tbody>
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<tr style="text-align:center;">
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<td><img src="images/triangle-5-12-13.svg" alt="5,12,13 Triangle" height="87" width="176"></td>
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<td style="width:30px;"> </td>
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<td><img src="images/triangle-9-40-41.svg" alt="9,40,41 Triangle" height="75" width="254"></td>
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</tr>
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<tr style="text-align:center;">
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<td>5,12,13 triangle</td>
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<td> </td>
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<td>9,40,41 Triangle</td>
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</tr>
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<tr style="text-align:center;">
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<td>5<sup>2</sup> + 12<sup>2</sup> = 13<sup>2</sup></td>
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<td> </td>
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<td>9<sup>2</sup> + 40<sup>2</sup> = 41<sup>2</sup></td>
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</tr>
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<tr style="text-align:center;">
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<td>25 + 144 = 169</td>
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<td> </td>
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<td>(try it yourself)</td>
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</tr>
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</tbody></table>
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<p>And there are infinitely many more.... read <a href="../pythagorean_triples.html">Pythagorean Triples</a> for more information.</p>
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<h2>But Wait ... There is More!</h2>
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<p>The 3,4,5 Triangle has a simple beauty. At its heart is a circle:</p>
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<p class="center"><img src="images/triangle-3-4-5-circle.svg" alt="3,4,5 Triangle" height="" width=""><br>The circle is a <a href="unit-circle.html">Unit Circle</a> (has a radius of exactly 1)</p>
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<p>And using that circle we can cut our triangle into 6 little triangles like this:</p>
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<p class="center"><img src="images/triangle-3-4-5-circ2.svg" alt="3,4,5 Triangle" height="" width=""><br>Each small triangle is also a right triangle!</p>
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<p>And notice that 1+2=3, 1+3=4 and 2+3=5, making the 3,4,5 triangle. So simple and neat.</p>
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<p><br></p>
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<p> </p>
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<div class="questions">2090,2091</div>
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<div class="related">
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<a href="../pythagoras.html">Pythagoras' Theorem</a>
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<a href="index.html">Geometry Index</a>
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</div>
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