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2011-02-01T00:00:01
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<p><span style="font-size: small;">Created with <a href="http://www.geogebra.org" target="_blank">GeoGebra</a></span></p>
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<div>Triangle ABC is any right angled triangle and X is a moveable point on the hypotenuse AC. The points P and Q are the feet of the perpendiculars from X to the sides of the triangle.</div>
<br></br>
<div>Find the position of X which makes the length of PQ a minimum.</div>
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<div class="framework">NOTES AND BACKGROUND<br></br>
<p>This dynamic image is drawn using Geogebra, free software and very easy to use. You can download your own copy of Geogebra from <a href="http://www.geogebra.org/cms/">http://www.geogebra.org/cms/</a> together with a good help manual and <a href="http://www.geogebra.org/cms/index.php?option=com_content&task=blogcategory&id=75&Itemid=61">Quickstart</a> for beginners.</p>
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<td align="center" valign="center"><mdo:image src="shrinkTri.gif" alt="Helpful diagram"></mdo:image></td>
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<p>The solution to this problem only uses two very simple
facts:</p>
<ol>
<li>The diagonals of a rectangle are equal in length.</li>
<li>For any triangle, the shortest distance from a vertex to the
opposite side is the perpendicular to that side.</li>
</ol>
<p>Putting the two together, for any right angled triangle the
length of <strong>P</strong> <strong>Q</strong> is a minimum when
<strong>X</strong> is the foot of the perpendicular from
<strong>A</strong> to <strong>B</strong> <strong>C</strong>. As the
point <strong>X</strong> moves along the hypotenuse the rectangle
<strong>A</strong> <strong>P</strong> <strong>X</strong>
<strong>Q</strong> changes but <strong>P</strong>
<strong>Q</strong> is always equal in length to <strong>A</strong>
<strong>X</strong>.</p>
<p>As a further challenge, you may like to prove that the position
of X is given by: ( <strong>C</strong> <strong>X</strong>/
<strong>X</strong> <strong>B</strong>) = ( <strong>C</strong>
<strong>A</strong>/ <strong>A</strong>
<strong>B</strong>)².</p>
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Shrink
X is a moveable point on the hypotenuse, and P and Q are the feet
of the perpendiculars from X to the sides of a right angled
triangle. What position of X makes the length of PQ a minimum?
2D Geometry, Shape and Space
Perpendicular lines
Sequences, Functions and Graphs
Maximise/minimise/optimise
2D Geometry, Shape and Space
Rectangles
Information and Communications Technology
Dynamic geometry