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<title>Mean Proportional and the Altitude and Leg Rules</title>
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<h1 class="center">Mean Proportional</h1>
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<p class="center"> <i>... and the <b>Altitude</b> and <b>Leg</b> Rules</i></p>
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<h2><span class="center">Mean Proportional</span></h2>
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<p>The mean proportional of <b>a</b> and <b>b</b> is the value <b>x</b> here:</p><p class="center larger"><span class="intbl"><em><span class="large">a</span></em><strong><span class="large">x</span></strong></span><span class="large"> = </span><span class="intbl"><em><span class="large">x</span></em><strong><span class="large">b</span></strong></span></p>
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<p class="center">"a is to x, as x is to b"</p>
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<p>It looks kind of hard to solve, doesn't it?</p>
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<p>But when we <a href="../algebra/cross-multiply.html">cross multiply</a> (multiply both sides by <b>b</b> and also by <b>x</b>) we get:</p>
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<table border="0" align="center">
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<tr>
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<td><p class="center larger"><span class="intbl"><em><span class="large">a</span></em><strong><span class="large">x</span></strong></span><span class="large"> = </span><span class="intbl"><em><span class="large">x</span></em><strong><span class="large">b</span></strong></span></p></td>
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<td width="90" align="center"><img src="../images/style/right-arrow.gif" width="46" height="46" alt="right arrow" /></td>
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<td><p class="center larger"><span class="intbl"><em><span class="large">ab</span></em><strong><span class="large">x</span></strong></span><span class="large"> = </span><span class="large">x</span></p></td>
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<td width="90" align="center"><img src="../images/style/right-arrow.gif" width="46" height="46" alt="right arrow" /></td>
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<td><span class="large">ab = x<sup>2</sup></span></td>
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</tr>
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</table>
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<p>And now we can solve for x:</p>
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<p class="center"><span class="large">x = √(ab)</span> </p>
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<div class="example">
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<h3>Example: What is the mean proportional of 2 and 18?</h3>
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<p>We are being asked "What is the value of x here?"</p>
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<p class="center larger"><span class="intbl"><em><span class="large">2</span></em><strong><span class="large">x</span></strong></span><span class="large"> = </span><span class="intbl"><em><span class="large">x</span></em><strong><span class="large">18</span></strong></span></p>
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<p class="center">"2 is to x, as x is to 18"</p>
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<p>We know how to solve it:</p>
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<p class="center larger">x = √(2×18) = √(36) = 6</p>
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<p>And this is what we end up with: </p>
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<p class="center larger"><span class="intbl"><em><span class="large">2</span></em><strong><span class="large">6</span></strong></span><span class="large"> = </span><span class="intbl"><em><span class="large">6</span></em><strong><span class="large">18</span></strong></span></p>
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</div>
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<p>It basically says that 6 is the <b>"multiplication</b> <b>middle</b>" (<b>2</b> times 3 is <b>6</b>, <b>6</b> times 3 is <b>18</b>)</p>
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<p class="center"><img src="images/mean-proportional-1.gif" width="188" height="58" alt="mean proportional 2 x3= 6 x3= 18" /></p>
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<p><i>(It is also the <a href="../numbers/geometric-mean.html">geometric mean</a> of the two numbers.) </i></p>
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<p>One more example so you get the idea:</p>
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<div class="example">
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<h3>Example: What is the mean proportional of 5 and 500? </h3>
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<p class="center larger">x = √(5×500)</p>
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<p class="center larger">x = √(2500) = <b>50</b></p>
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<p>So it is like this:</p>
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<p class="center"><img src="images/mean-proportional-2.gif" width="198" height="53" alt="mean proportional 5 x10= 50 x10= 500" /></p>
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</div><p> </p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/mean-proportional-3.svg" alt="mean proportional similar triangles inside" /></p>
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<h2>Right Angled Triangles</h2>
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<p>We can use the mean proportional with <a href="../right_angle_triangle.html">right angled triangles</a>.</p>
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<p>First, an interesting thing:</p>
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<ul>
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<li>Take a right angled triangle <b>sitting on its hypotenuse</b> (long side)</li>
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<li>Put in an altitude line</li>
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<li>It divides the triangle into two other triangles, yes?</li>
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</ul>
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<p>Those two new triangles are <a href="similar.html">similar</a> to each other, and to the original triangle!</p>
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<p>This is because they all have the same three angles. </p>
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<p>Try it yourself: cut a right angled triangle from a piece of paper, then cut it through the altitude and see if the pieces are really similar.</p>
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<p>We can use this knowledge to solve some things.</p>
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<p>In fact we get two rules:</p>
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<h2>Altitude Rule</h2>
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<p>The altitude is the mean proportional between the left and right parts of the hyptonuse, like this:</p>
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<p class="center"><img src="images/mean-proportional-4.svg" alt="mean proportional left/altitude = altitude/right" /></p>
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<div class="example">
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<h3>Example: Find the height <b>h</b> of the altitude (AD)</h3>
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<p class="center"><img src="images/mean-proportional-8.gif" width="230" height="124" alt="mean proportional 4.9 h 10" /></p>
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<p>Use the Altitude Rule: </p>
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<p class="center larger"><span class="intbl"><em>left</em><strong>altitude</strong></span> = <span class="intbl"><em>altitude</em><strong>right</strong></span></p>
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<p>Which for us is:</p>
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<p class="center larger"><span class="intbl"><em>4.9</em><strong>h</strong></span> = <span class="intbl"><em>h</em><strong>10</strong></span></p>
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<p>And solve for h:</p>
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<div class="so">h<sup>2</sup> = 4.9 × 10 = 49</div>
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<div class="so">h = √49 = 7</div>
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</div>
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<h2>Leg Rule</h2>
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<p>Each leg of the triangle is the mean proportional between the <b>hypotenuse</b> and the <b>part of the hypotenuse directly below the leg</b>: </p>
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<table border="0" align="center">
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<tr>
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<td><img src="images/mean-proportional-5.svg" alt="mean proportional hyp/leg = leg/part" /></td>
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<td> and </td>
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<td><img src="images/mean-proportional-6.svg" alt="mean proportional hyp/leg = leg/part" /></td>
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</tr>
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</table>
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<div class="example">
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<h3>Example: What is <b>x</b> (the length of leg AB) ?</h3>
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<p class="center"><img src="images/mean-proportional-7.gif" width="249" height="157" alt="mean proportional x 9 7" /></p>
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<p>First find the hypotenuse: BC = BD + DC = 9 + 7 = 16</p>
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<p>Now use the Leg Rule:</p>
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<p class="center larger"><span class="intbl"><em>hypotenuse</em><strong>leg</strong></span> = <span class="intbl"><em>leg</em><strong>part</strong></span></p>
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<p>Which for us is:</p>
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<p class="center larger"><span class="intbl"><em>16</em><strong>x</strong></span> = <span class="intbl"><em>x</em><strong>9</strong></span></p>
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<p>And solve for x:</p>
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<div class="so">x<sup>2</sup> = 16 × 9 = 144</div>
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<div class="so">x = √144 = 12 </div>
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</div>
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<p>Here is a real world example: </p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/mean-proportional-9.gif" width="216" height="235" alt="mean proportional kite PO is 80, OR is 180" /></p>
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<h3>Example: Sam loves kites!</h3>
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<p> </p>
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<p>Sam wants to make a really big kite:</p>
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<ul>
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<li>It has two struts PR and QS that intersect at a right angle at O. </li>
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<li>PO = 80 cm and OR = 180 cm. </li>
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<li>The fabric of the kite has right angles at Q and S.</li>
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</ul>
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<p> </p>
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<p>Sam wants to know the length for the strut QS, and also the lengths of each side.</p>
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<p>We only need to look at half the kite to do the calculations. Here is the left half rotated 90°</p>
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<p class="center"><img src="images/mean-proportional-10.gif" width="216" height="119" alt="mean proportional triangle p, r, h, 180 and 80" /></p>
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<p>Use the altitude rule to find <b>h</b>:</p>
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<div class="so">h<sup>2</sup> = 180 × 80 = 14400</div>
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<div class="so">h = √14400 = 120 cm</div>
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<p>So the full length of the strut QS = 2 × 120 cm = <b>240 cm </b></p>
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<p> </p>
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<p>The length RP = RO + OP = 180 cm + 80 cm = <b>260 cm</b> </p>
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<p>Now use the Leg Rule to find <b>r</b> (leg QP):</p>
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<div class="so">r<sup>2</sup> = 260 × 80 = 20800</div>
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<div class="so">r = √20800 = <b>144 cm</b> to nearest cm</div>
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<p> </p>
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<p> Use the Leg Rule again to find <b>p</b> (leg QR):</p>
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<div class="so">p<sup>2</sup> = 260 × 180 = 46800</div>
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<div class="so">p = √46800 = <b>216 cm</b> to nearest cm</div>
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<p> </p>
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<p>Tell Sam the strut QS will be <b>240 cm</b>, and the sides will be <b>144 cm</b> and <b>216 cm</b>.</p>
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<p>Can't wait for a windy day!</p>
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</div>
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<p> </p>
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<script type="text/javascript">getQ(7551, 7552, 7553, 7554, 7555, 7556, 7557, 7558, 7559, 7560);</script>
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<div class="related"><a href="../algebra/cross-multiply.html">Cross Multiply</a>
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<a href="index.html">Geometry Index</a> <a href="../algebra/trigonometry-index.html">Trigonometry Index</a> </div>
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