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<!-- #BeginEditable "doctitle" -->
<title>Similar</title>
<meta name="description" content="Learn what makes two shapes Similar (hint: it has to do with resizing!)" />
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<h1 align="center">Similar</h1><p style="float:left; margin: 0 10px 5px 0;">&nbsp;</p>
<div class="def">
<span style="float:left; margin: 0 10px 5px 0;"><img src="images/similar-resize.svg" alt="similar triangles" /></span>
<p>Two shapes are <b>Similar</b> when one can become the other after a <b>resize</b>, flip, slide or turn.</p>
</div>
<h2>Resizing</h2>
<p>If one shape can become another using <a href="resizing.html">Resizing</a> (also called<b> </b><i>dilation, contraction, compression, enlargement</i> or even <i>expansion</i>), then the shapes are <b>Similar</b>:</p>
<table border="0" align="center">
<tr>
<td><img src="images/resize.svg" alt="resize" /></td>
<th>These Shapes are Similar!</th>
</tr>
</table>
<p>If there is no need to resize, then the shapes are better called <a href="congruent.html">Congruent</a>*.</p>
<h2>There may be Turns, Flips or Slides, Too!</h2>
<p> Sometimes it can be hard to see if two shapes are Similar, because you may also need to turn, flip or slide a shape.</p>
<table border="0" align="center">
<tr>
<th><a href="rotation.html">Rotation</a></th>
<td align="center"><img src="images/rotation-2d.svg" alt="rotation 2d" /></td>
<th>Turn!</th>
</tr>
<tr>
<th><a href="reflection.html">Reflection</a></th>
<td align="center"><img src="images/reflect-graph.svg" alt="reflect on graph" /></td>
<th>Flip!</th>
</tr>
<tr>
<th><a href="translation.html">Translation</a></th>
<td align="center"><img src="images/translation.svg" alt="translation on graph" /></td>
<th>Slide!</th>
</tr>
</table>
<h2>Examples</h2>
<p align="left">Here are 3 examples of shapes that are <b>Similar:</b> </p>
<table border="0" align="center">
<tr align="center">
<td><img src="images/similar-resize.svg" style="max-width: 100%;" alt="similar triangles" /></td>
<td width="50">&nbsp;</td>
<td><img src="images/similar-resize-flip.svg" style="max-width: 100%;" alt="similar resize flip" /></td>
<td width="50">&nbsp;</td>
<td><img src="images/similar-resize-rotate.svg" style="max-width: 100%;" alt="similar resize rotate" /></td>
</tr>
<tr align="center">
<td>Resized</td>
<td>&nbsp;</td>
<td>Resized and<br>
Reflected</td>
<td>&nbsp;</td>
<td>Resized and<br>
Rotated</td>
</tr>
</table>
<h2>Why is it Useful?</h2>
<p>When two shapes are similar, then:</p>
<ul>
<li>corresponding angles are equal, and</li>
<li>the lines are in proportion.</li>
</ul>
<p>This can make life a lot easier when solving geometry puzzles, as in this example:</p>
<div class="example">
<h3>Example: What is the missing length here? </h3>
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/similar-triangles.svg" alt="similar triangles" /></p>
<p>&nbsp;</p>
<p>Notice that the red triangle has the <b>same angles</b> as the blue triangle ...</p>
<p>... they both have one <b>right angle</b>, and a <b>shared angle</b> in the left corner</p>
<p>&nbsp;</p>
<p>In fact we can flip the red triangle over, rotate it a little, resize it, and it will fit exactly on top of the blue triangle. So they are <b>similar triangles</b>.</p>
<p>So the line lengths are in proportion:</p>
<ul>
<li>The blue triangle has two sides with the ratio 130/127</li>
<li> The red triangle has <b>matching</b> sides in the ratio ?/80</li>
</ul>
<p>and we can calculate:</p>
<h3 align="center">? = 80 &times; <span class="intbl"><em>130</em><strong>127</strong></span> = 81.9</h3>
<p align="center">(No fancy calculations, just common sense.)</p>
</div>
<h2>Congruent or Similar?</h2>
<p>Shapes are <a href="congruent.html">Congruent</a> when they are the same size (but may have been rotated, reflected or moved). So when the shapes become the same:</p>
<div class="simple">
<table border="0" align="center">
<tr>
<th>When we ...</th>
<th>&nbsp;</th>
<th>Then the shapes are ...</th>
</tr>
<tr>
<td>... only Rotate, Reflect and/or Translate&nbsp;</td>
<td><img src="../images/style/right-arrow.gif" width="46" height="46" alt="right arrow" /></td>
<td>
<h3 align="center">Congruent</h3> </td>
</tr>
<tr>
<td>... also need to <b>Resize</b></td>
<td><img src="../images/style/right-arrow.gif" width="46" height="46" alt="right arrow" /></td>
<td>
<h3 align="center">Similar</h3> </td>
</tr>
</table>
<br />
</div>
<h2>*Are Congruent Shapes also Similar? </h2>
<p>Most people (including us) say &quot;<b>Congruent</b> shapes are also <b>Similar</b>&quot;.</p>
<div class="example">
<h3>Example: </h3>
<p style="float:left; margin: 15px;"><img src="images/congruent-turn.svg" alt="congruent turn" /></p>
<p>We can move and rotate the orange shape to exactly match the blue shape, so the two shapes are <b>Congruent</b>. </p>
<p>We <b>don't have to</b> resize for the shapes to be similar! So they are <b>also Similar</b> even though no resizing was needed. </p>
</div>
<p>&nbsp;</p>
<div class="questions">
<script type="text/javascript">getQ(781, 3272, 6050, 6051, 6052, 3273, 3274, 3275, 631, 632, 1802, 1803);</script>&nbsp;
</div>
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<a href="congruent.html">Congruent</a>
<a href="transformations.html">Transformations</a>
<a href="rotation.html">Rotation</a>
<a href="reflection.html">Reflection</a>
<a href="translation.html">Translation</a>
<a href="resizing.html">Resizing</a>
<a href="symmetry.html">Symmetry Index</a>
<a href="index.html">Geometry Index</a></div>
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