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145 lines
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HTML
145 lines
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HTML
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<title>Similar Triangles</title>
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<h1 align="center">Similar Triangles</h1>
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<p>Two triangles are <a href="similar.html">Similar</a> if the only difference is size (and possibly the need to turn or flip one around).</p>
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<p align="center">These triangles are all similar:</p><p align="center"><img src="images/tri-similar1.svg" alt="triangles similar different sizes and rotations" /></p>
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<p align="center">(Equal angles have been marked with the same number of arcs) </p>
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<p>Some of them have different sizes and some of them have been turned or flipped. </p>
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<p> </p>
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<p>For similar triangles:</p>
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<p class="center"><img src="images/tri-corresponding-angles.svg" alt="corresponding angles on two triangles" /><br>
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All corresponding angles are equal</p>
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<p class="center"><i>and</i></p>
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<p class="center"><img src="images/tri-corresponding-sides.svg" alt="corresponding sides on two triangles" /><br>
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All corresponding sides have the same <a href="../numbers/ratio.html">ratio</a><br>
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</p>
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<p>Also notice that the <span class="center">corresponding sides</span> face the corresponding angles. For example the sides that face the angles with two arcs are corresponding.</p>
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<h2>Corresponding Sides </h2>
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<p>
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In similar triangles, <span class="center">corresponding</span> sides are always in the same ratio. </p><p>For example:</p><p align="center"><img src="images/tri-similar2.gif" width="360" height="133" alt="triangles similar R: (6,7,8) and S: (b,a,6.4)" /></p>
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<p>Triangles <b>R</b> and <b>S </b>are similar. The equal angles are marked with the same numbers of arcs. </p>
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<p>What are the corresponding lengths?</p>
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<ul>
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<li>The lengths <b>7</b> and <b>a</b> are corresponding (they face the angle marked with one arc)</li>
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<li>The lengths <b>8</b> and <b>6.4</b> are corresponding (they face the angle marked with two arcs) </li>
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<li>The lengths <b>6</b> and <b>b</b> are corresponding (they face the angle marked with three arcs)</li>
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</ul>
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<h2>Calculating the Lengths of Corresponding Sides</h2>
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<p><b></b>We can sometimes calculate lengths we don't know yet. </p>
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<ul>
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<li>Step 1: Find <b>the ratio</b> of corresponding sides</li>
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<li>Step 2: Use that ratio to find the unknown lengths</li>
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</ul>
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<div class="example">
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<h3>Example: Find lengths a and b of Triangle S</h3>
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<p class="center"><img src="images/tri-similar2.gif" width="360" height="133" alt="triangles similar R: (6,7,8) and S: (b,a,6.4)" /></p>
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<h3>Step 1: Find the ratio</h3>
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<p class="indent50px">We know all the sides in Triangle<b> R</b>, and <br />
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We know the side <b>6.4</b> in Triangle<b> S</b></p>
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<p class="indent50px">The 6.4 faces the angle marked with two arcs as does the side of length 8 in triangle <b>R</b>.</p>
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<p class="indent50px">So we can match <b>6.4 with 8</b>, and so the <a href="../numbers/ratio.html">ratio</a> of sides in triangle<b> S</b> to triangle <b>R </b>is:</p>
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<p align="center" class="indent50px"> 6.4 to 8 </p>
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<p class="indent50px">Now we know that the lengths of sides in triangle<b> S</b> are all <b>6.4/8 times</b> the lengths of sides in triangle <b>R</b>.</p>
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<h3>Step 2: Use the ratio</h3>
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<div class="indent50px">
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<p><b>a</b> faces the angle with one arc as does the side of length 7 in triangle <b>R</b>.</p>
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<div class="so"><b>a</b> = (6.4/8) × 7 = 5.6</div>
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<p> </p>
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<p><b>b</b> faces the angle with three arcs as does the side of length 6 in triangle <b>R</b>.</p>
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<div class="so"><b>b</b> = (6.4/8) × 6 = 4.8</div>
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<p> </p>
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<p>Done! </p>
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</div>
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</div>
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<div class="fun">
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<p><span style="float:left; margin: 0 20px 5px 0;"><img src="../measure/images/thumb-distance-far.gif" width="360" height="144" alt="thumb distance far" /></span></p>
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<p>Did You Know?</p>
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<p>Similar triangles can help you <a href="../measure/estimate-distance.html">estimate distances</a>.</p>
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</div>
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<div style="clear:both"></div>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(736, 737, 738, 1526, 1527, 1528, 3951, 3952, 3953, 3954);</script>
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</div>
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<div class="related">
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<a href="similar.html">Similar</a>
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<a href="triangles-similar-finding.html">Finding Similar Triangles</a>
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<a href="triangles-similar-theorems.html">Similar Triangle Theorems</a>
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<a href="congruent.html">Congruent</a>
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<a href="triangles-congruent.html">Congruent Triangles</a>
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<a href="../algebra/trigonometry-index.html">Trigonometry Index</a> </div>
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