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180 lines
8.4 KiB
HTML
180 lines
8.4 KiB
HTML
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<h1 class="center">Parallel and Perpendicular Lines and Planes</h1>
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<p class="center"><br />
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</p>
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<div class="simple">
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<table border="0" align="center">
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<tr>
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<td>This is a <a href="line.html">line</a>:</td>
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<td><span class="center"><img src="images/line.svg" alt="line" /></span></td>
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</tr>
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</table>
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</div>
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<p class="center">Well it is an <i>illustration</i> of a line, <br />
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because a line has <b>no thickness</b>, and <b>no ends</b> (goes on forever). </p>
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<p class="center"> </p>
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<div class="simple">
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<table border="0" align="center">
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<tr>
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<td>This is a <a href="plane.html">plane</a>:<br /> </td>
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<td><span class="center"><img src="images/plane.svg" alt="plane" /></span></td>
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</tr>
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</table>
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</div> <p class="center">
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OK, an <i>illustration</i> of a plane, <br>
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because a plane is a <b>flat surface</b> with <b>no thickness</b> that <b>extends forever</b>.
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<br />
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(But here we draw edges just to make the illustrations clearer.)</p>
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<h2>Perpendicular Lines</h2>
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<p>Two line are perpendicular when they are at right angles to each other.</p>
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<p>The red line is perpendicular to the blue line in each of these examples:</p>
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<p class="center"><img src="images/perpendicular.svg" alt="Perpendicular Example" /></p>
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<p>(Read more about <a href="../perpendicular-parallel.html">perpendicular lines</a>.)</p>
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<p> </p>
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<h2>Perpendicular to a Plane</h2>
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<p>A line is perpendicular to a plane when it extends directly away from it, like a pencil standing up on a table.</p>
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<p><img src="images/plane-perp-photo1.jpg" width="250" height="187" alt="plane perpendicular photo" /></p>
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<p>If the pencil is perpendicular to a line on the table, then it <b>might be</b> perpendicular to the table:</p>
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<p><img src="images/plane-perp-photo2.jpg" width="250" height="188" alt="plane perpendicular photo" /></p>
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<p>Or it <b>might be leaning over</b>:</p>
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<p><img src="images/plane-perp-photo3.jpg" width="250" height="189" alt="plane perpendicular photo" /></p>
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<p>But when it is perpendicular to <b>two lines</b> (where they intersect) then it <b>is </b>perpendicular to the table:</p>
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<p><img src="images/plane-perp-photo4.jpg" width="250" height="188" alt="plane perpendicular photo" /></p>
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<p>It can't point anywhere else but directly away from the table.</p>
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<p>So we can say this:</p>
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<table width="100%" border="0">
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<tr>
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<td><img src="images/plane-perp-1.gif" width="273" height="142" alt="plane perpendicular" /></td>
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<td><p>When a line is perpendicular to<b> two lines on the plane</b> (where they intersect), it is <b>perpendicular to the plane</b>.</p></td>
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</tr>
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<tr>
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<td><img src="images/plane-perp-1a.gif" width="272" height="130" alt="plane perpendicular" /></td>
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<td><p>It will also be perpendicular to all lines on the plane that intersect there.</p></td>
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</tr>
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</table>
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<p>And there is a lot more we can say:</p>
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<table border="0">
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<tr>
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<td><img src="images/plane-perp-2.gif" width="272" height="143" alt="plane perpendicular" /></td>
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<td> <p> Through a given point there passes:</p>
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<ul>
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<li>one and only one <b>line perpendicular to a plane</b></li>
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<li>one and only one <b>plane perpendicular to a line</b></li>
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</ul> </td>
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</tr>
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</table>
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<p> </p>
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<table width="100%" border="0">
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<tr>
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<td><img src="images/plane-perp-3.gif" width="270" height="177" alt="plane perpendicular" /></td>
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<td>two lines perpendicular to the same plane are coplanar (they lie on the same plane - the orange one) </td>
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</tr>
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</table>
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<table width="100%" border="0">
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<tr>
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<td align="right">If that is a little hard to understand, imagine two pencils standing on a table: they are in the same plane (the piece of cardboard):</td>
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<td> </td>
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<td><img src="images/plane-perp-photo6a.jpg" width="200" height="187" alt="plane perpendicular photo" /></td>
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</tr>
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</table>
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<h2>Perpendicular Planes</h2>
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<table width="100%" border="0">
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<tr>
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<td><img src="images/plane-perp-4.gif" width="275" height="201" alt="plane perpendicular" /></td>
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<td>A plane is perpendicular to another plane when it has a line that is perpendicular to the other plane </td>
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</tr>
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<tr>
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<td><img src="images/plane-perp-5.gif" width="268" height="198" alt="plane perpendicular" /></td>
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<td>And when a line is perpendicular to a plane, then every plane containing the line is perpendicular to that plane </td>
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</tr>
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</table>
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<h2>Parallel Planes</h2>
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<table width="100%" border="0">
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<tr>
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<td><img src="images/plane-perp-7.gif" width="276" height="257" alt="plane perpendicular" /></td>
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<td>When two planes are perpendicular to the same line, they are parallel planes</td>
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</tr>
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</table>
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<p> </p>
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<table width="100%" border="0">
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<tr>
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<td><img src="images/plane-perp-6.gif" width="272" height="209" alt="plane perpendicular" /></td>
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<td>When a plane intersects <b>two parallel planes</b>, the intersection is <b>two parallel lines.</b></td>
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</tr>
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</table>
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<p> </p>
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<p> </p>
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<div class="related">
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<a href="../perpendicular-parallel.html">Parallel and Perpendicular Lines</a><a href="index.html">Geometry Index</a>
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