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<h1 class="center">Pythagorean Theorem Algebra Proof</h1>
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<h2>What is the Pythagorean Theorem?</h2>
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<p>You can learn all about the <a href="../pythagoras.html">Pythagorean Theorem</a>, but here is a quick summary:</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/triangle-abc.svg" alt="triangle abc"></p>
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<p>The Pythagorean Theorem says that, <i>in a right triangle,</i> the <a href="../square-root.html">square</a> of <b>a</b> (which is a×a, and is written <b>a<sup>2</sup></b>) plus the square of <b>b</b> (<b>b<sup>2</sup></b>) is equal to the square of <b>c</b> (<b>c<sup>2</sup></b>):</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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<h2>Proof of the Pythagorean Theorem using Algebra</h2>
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<p>We can show that <b>a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></b> using <a href="../algebra/index.html">Algebra</a></p>
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<p>Take a look at this diagram ... it has that "abc" triangle in it (four of them actually):</p>
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<div class="center"><img src="images/pythagorean-theorem-proof.png" alt="Squares and Triangles" height="236" width="233"> </div>
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<h2>Area of Whole Square</h2>
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<p>It is a big square, with each side having a length of <b>a+b</b>, so the <b>total area</b> is:</p>
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<p class="center larger">A = (a+b)(a+b)</p>
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<h2>Area of The Pieces</h2>
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<p>Now let's add up the areas of all the smaller pieces:</p>
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<div class="tbl">
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<div class="row"><span class="left">First, the smaller (tilted) square
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has an area of: </span><span class="right"><span class="large"> c<sup>2</sup></span></span></div><br>
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<div class="row"><span class="left">Each of the four triangles has an area of:</span><span class="right"><span class="large"><span class="intbl"><em>ab</em><strong>2</strong></span></span></span></div>
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<div class="row"><span class="left">So all four of them together is:</span><span class="right"><span class="large"><span class="intbl"><em>4ab</em><strong>2</strong></span> = 2ab</span></span></div><br>
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<div class="row"><span class="left">Adding up the tilted square and the 4 triangles gives:</span><span class="right"><span class="large">A = c<sup>2</sup> + 2ab</span></span></div>
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</div>
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<h2>Both Areas Must Be Equal</h2>
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<p>The area of the <b>large square</b> is equal to the area of the <b>tilted square and the 4 triangles</b>. This can be written as:</p>
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<p class="center larger">(a+b)(a+b) = c<sup>2</sup> + 2ab</p>
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<p>NOW, let us rearrange this to see if we can get the pythagoras theorem:</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+b)(a+b) = c<sup>2</sup> + 2ab</span></div>
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<div class="row"><span class="left">Expand (a+b)(a+b):</span><span class="right">a<sup>2</sup> + 2ab + b<sup>2</sup> = c<sup>2</sup> + 2ab</span></div>
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<div class="row"><span class="left">Subtract "2ab" from both sides:</span><span class="right">a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></span></div>
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</div>
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<p class="center large">DONE!</p>
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<p>Now we can see why the Pythagorean Theorem works ... and it is actually a <b>proof</b> of the Pythagorean Theorem.</p>
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<p class="center larger">This proof came from China over 2000 years ago!</p>
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<p>There are many more proofs of the Pythagorean theorem, but this one works nicely.</p>
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<p> </p>
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<div class="related">
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<a href="../pythagoras.html">Pythagorean Theorem</a>
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