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155 lines
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HTML
155 lines
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HTML
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<title>Generalizations of the Pythagorean Theorem</title>
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<meta name="description" content="Learn the generalized form of the Pythagorean Theorem">
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1>Generalizations of Pythagoras' Theorem</h1>
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<h2>Pythagoras' Theorem</h2>
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<p>Let's start with a quick refresher of the traditional well-known
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Pythagoras' Theorem.</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/triangle-abc.svg" alt="triangle abc" style="width: 201px; height: 115px;"></p>
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<p style="text-align: center;">Pythagoras' Theorem says that, in a
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right angled triangle:<br>
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the square of the hypotenuse (<strong>c</strong>) is equal to the sum of
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the squares of the other two sides (<strong>a</strong> and <strong>b</strong>).</p>
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<p class="center large">a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></p>
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<p>You can learn more about <a href="../pythagoras.html">Pythagoras'
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Theorem</a> and review its <a href="pythagorean-theorem-proof.html">algebraic
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proof</a>.</p>
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<h2>Pythagoras' Theorem in 3D</h2>
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<p>The world we live in has three <a href="dimensions.html">dimensions</a>,
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so what would happen if we consider the <a href="pythagoras-3d.html">Pythagorean
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Theorem in 3D</a>?</p>
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<p>Well, the Theorem still holds, and we would have something like this:</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/pythagoras-3d-d.svg" alt="Pythagoras 3D" style="width: 252px; height: 198px;"></p>
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<p style="text-align: center;"><br></p>
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<p style="text-align: center;">The square of the distance <strong>c</strong>
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from the bottom-most left front corner to the top-most right back corner
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of this cuboid whose sides are <strong>x</strong>, <strong>y</strong>
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and <strong>z</strong>, is:</p>
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<p class="center large">c<sup>2</sup> = x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup></p>
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<p><br></p>
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<p>And this is part of a pattern that extends onwards into any number of
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dimensions. For the n-th dimension, we have:</p>
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<p class="center large">c<sup>2</sup> = a<sub>1</sub><sup>2</sup> + a<sub>2</sub><sup>2</sup>
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+ ... + a<sub>n</sub><sup>2</sup></p>
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<p>So we can generalize Pythagoras' Theorem, going from 2D
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to 3D and up until any number of dimensions.</p>
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<h2>Law of Cosines</h2>
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<p>What if the triangle does not have a right angle?</p>
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For any triangle:
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<p style="float:left; margin: 0 10px 5px 0;"><img src="../algebra/images/triangle-sides-angles.svg" alt="triangle angles A, B, C and sides a, b, c" style="width: 265px; height: 152px;"></p>
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<p style="text-align: center;"><b>a</b>, <b>b</b> and <b>c</b> are
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sides. <b><br>
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C</b> is the angle opposite to side c<br>
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<br>
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<b>The Law of Cosines</b> (also called the <b>Cosine Rule</b>) says:</p>
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<p class="center large">c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup> −
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2ab cos(C)</p>
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<p><br></p>
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<p>It has <b>a<sup>2</sup></b>, <b>b<sup>2</sup></b> and <b>c<sup>2</sup></b>, and an extra term: <b>2ab cos(C)<br>
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</b></p>
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<p>Learn how to use it and find out more at <a href="../algebra/trig-cosine-law.html">Law
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of Cosines</a>!</p>
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<p><br></p>
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<p>These two generalizations are already nice and inspiring... But wait,
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there is more!</p>
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<h2>Pythagoras' Theorem and Areas</h2>
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<p>Do they need to be squares on the triangle's sides?</p>
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<p>What about semicircles?</p>
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<p class="center"><img src="images/pythagoras-circle.svg" alt="Pythagoras semicircle"></p>
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<p>Read more at <a href="pythagoras-areas.html">Pythagoras' Theorem and Areas</a>.</p>
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<h2>Higher Exponents?</h2>
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<p>Finally, another type of generalization is to try higher exponents:</p>
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<p class="center large">a<sup>n</sup> + b<sup>n</sup>
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= c<sup>n</sup> <i>n>2</i></p>
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<p>An example is <b>n=3</b>: are there any whole numbers that make this true?</p>
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<p class="center large">a<sup>3</sup> + b<sup>3</sup> = c<sup>3</sup></p>
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<p>In geometry this is the same as asking: </p>
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<div class="fun">
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<p class="center larger">Using only integer sides, can we split a cube
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into two cubes?</p>
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</div>
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<p><b>Can we?</b> Your Turn! To answer this, search the web for the well-known mathematician
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Pierre Fermat and his famous Last Theorem.</p>
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<p><br></p>
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<div class="related">
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<a href="../pythagoras.html">Pythagorean Theorem</a>
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<a href="pythagorean-theorem-proof.html">Pythagorean Theorem Algebraic Proof</a>
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<a href="pythagoras-3d.html">Pythagorean Theorem in 3D</a>
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<a href="index.html">Geometry Index</a>
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