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166 lines
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<title>Theorems about Similar Triangles</title>
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<h1 align="center">Theorems about Similar Triangles</h1>
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<h2>1. The Side-Splitter Theorem</h2><p align="center">
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<img src="images/tri-similar6.gif" width="226" height="208" alt="triangles similar ABC and ADE" />
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</p>
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<div class="def">
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<p>If ADE is any triangle and BC is drawn parallel to DE, then <span class="intbl"><em>AB</em><strong>BD</strong></span> = <span class="intbl"><em>AC</em><strong>CE</strong></span></p>
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</div>
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<p>
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To show this is true, draw the line BF parallel to AE to complete a parallelogram BCEF:</p><p align="center"><img src="images/tri-similar7a.gif" width="223" height="201" alt="triangles similar ABC and ADE: BF and EC same" /></p>
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<p>Triangles ABC and BDF have exactly the same angles and so are similar (Why? See the section called <b> AA</b> on the page <a href="triangles-similar-finding.html">How To Find if Triangles are Similar</a>.)</p>
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<ul>
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<li>Side AB corresponds to side BD and side AC corresponds to side BF.</li>
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<li>So AB/BD = AC/BF</li>
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<li>But BF = CE</li>
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<li>So AB/BD = AC/CE</li>
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</ul>
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<h2>The Angle Bisector Theorem</h2>
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<p align="center">
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<img src="images/tri-similar8.gif" width="304" height="120" alt="triangles similar ABC point D" /></p>
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<div class="def">
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<p>If ABC is any triangle and AD bisects (cuts in half) the angle BAC, then <span class="intbl"><em>AB</em><strong>BD</strong></span> = <span class="intbl"><em>AC</em><strong>DC</strong></span></p>
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</div>
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<p>To show this is true, we can label the triangle like this:</p>
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<p align="center"><img src="images/tri-similar9.gif" width="300" height="113" alt="triangles similar angles x and x at A and angles y and 180-y at D" /></p>
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<ul>
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<li>Angle <span class="times"></span>BAD = Angle DAC = x° </li>
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<li>Angle ADB = y° </li>
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<li>Angle ADC = (180−y)°</li>
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</ul>
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<div class="tbl">
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<div class="row"><span class="left">By the <a href="../algebra/trig-sine-law.html">Law of Sines</a> in triangle ABD:</span><span class="right"><span class="intbl"><em>sin(x)</em><strong>BD</strong></span> = <span class="intbl"><em>sin(y)</em><strong>AB</strong></span></span></div>
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<div class="row"><span class="left">Multiply both sides by AB:</span><span class="right"><span class="intbl"><em>sin(x)AB </em><strong>BD</strong></span> = <span class="intbl"><em>sin(y)</em><strong>1</strong></span></span></div>
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<div class="row"><span class="left">Divide both sides by sin(x):</span><span class="right"><b><span class="intbl"><em>AB</em><strong>BD</strong></span> = <span class="intbl"><em>sin(y)</em><strong>sin(x)</strong></span></b></span></div>
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<div> </div>
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<div class="row"><span class="left">By the Law of Sines in triangle ACD:</span><span class="right"><span class="intbl"><em>sin(x)</em><strong>DC</strong></span> = <span class="intbl"><em>sin(180−y)</em><strong>AC</strong></span></span></div>
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<div class="row"><span class="left">Multiply both sides by AC:</span><span class="right"><span class="intbl"><em>sin(x)AC</em><strong>DC</strong></span> = <span class="intbl"><em>sin(180−y)</em><strong>1</strong></span></span></div>
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<div class="row"><span class="left">Divide both sides by sin(x):</span><span class="right"><span class="intbl"><em>AC</em><strong>DC</strong></span> = <span class="intbl"><em>sin(180−y)</em><strong>sin(x)</strong></span></span></div>
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<div class="row"><span class="left">But <b>sin(180−y) = sin(y)</b>:</span><span class="right"><b><span class="intbl"><em>AC</em><strong>DC</strong></span> = <span class="intbl"><em>sin(y)</em><strong>sin(x)</strong></span></b></span></div>
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</div>
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<p>Both <span class="intbl"><em>AB</em><strong>BD</strong></span> and <span class="intbl"><em>AC</em><strong>DC</strong></span> are equal to <span class="intbl"><em>sin(y)</em><strong>sin(x)</strong></span>, so:</p>
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<p class="center large"><span class="intbl"><em>AB</em><strong>BD</strong></span> = <span class="intbl"><em>AC</em><strong>DC</strong></span></p>
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<p>In particular, if triangle ABC is isosceles, then triangles ABD and ACD are <a href="triangles-congruent.html">congruent triangles</a></p><p align="center"><img src="images/tri-similar10.gif" width="335" height="155" alt="triangles similar right angle at D" /></p>
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<p>And the same result is true:</p><p class="center large"><span class="intbl"><em>AB</em><strong>BD</strong></span> = <span class="intbl"><em>AC</em><strong>DC</strong></span></p>
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<h2>3. Area and Similarity</h2>
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<div class="def">
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<p align="center">If two similar triangles have sides in the ratio x:y, <br />
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<br />
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then their areas are in the ratio x<sup>2</sup>:y<sup>2</sup></p>
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</div>
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<h3> Example:</h3>
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<p>These two triangles are similar with sides in the ratio 2:1 (the sides of one are twice as long as the other): </p>
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<p align="center"> <img src="images/tri-similar12a.gif" width="320" height="111" alt="triangles similar large and small" /></p>
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<p>What can we say about their areas?</p>
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<p>The answer is simple if we just draw in three more lines:</p><p align="center"><img src="images/tri-similar12b.gif" alt="triangles similar small fits inside large 3 times" /></p>
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<p>We can see that the small triangle fits into the big triangle <b>four times</b>.</p>
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<p>So when the lengths are <b>twice</b> as long, the area is <b>four times</b> as big</p>
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<p>So the ratio of their areas is 4:1 </p>
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<p>We can also write 4:1 as <span class="large">2<sup>2</sup>:1</span></p>
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<h3>The General Case:</h3>
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<p align="center"><img src="images/tri-similar11.gif" alt="triangles similar ABC and PQR" /></p>
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<p align="center" class="larger">Triangles ABC and PQR are similar and have sides in the ratio <b>x:y</b> </p>
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<p>We can find the areas using this formula from <a href="../algebra/trig-area-triangle-without-right-angle.html">Area of a Triangle</a>:</p>
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<p align="center" class="large">Area of ABC = <span class="intbl"><em>1</em><strong>2</strong></span>bc sin(A)</p>
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<p align="center" class="large">Area of PQR = <span class="intbl"><em>1</em><strong>2</strong></span>qr sin(P)</p>
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<p>And we know the lengths of the triangles are in the ratio <b>x:y</b></p>
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<p align="center"> q/b = y/x, so: <b>q = by/x</b></p>
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<p align="center">and r/c = y/x, so <b>r = cy/x</b></p>
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<p>Also, since the triangles are similar, <b>angles A and P</b> are the same:</p>
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<p align="center"><b>A = P</b></p>
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<p>We can now do some calculations:</p><div class="tbl">
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<div class="row"><span class="left"><b>Area of triangle PQR</b>:</span><span class="right"><b><span class="intbl"><em>1</em><strong>2</strong></span>qr sin(P)</b></span></div>
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<div class="row"><span class="left">Put in "q = by/x", "r = cy/x" and "P=A":</span><span class="right"><span class="intbl"><em>1</em><strong>2</strong></span><span class="intbl"><em>(by)(cy) sin(A)</em><strong>(x)(x)</strong></span></span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right"><span class="intbl"><em>1</em><strong>2</strong></span><span class="intbl"><em>bcy<sup>2</sup> sin(A)</em><strong>x<sup>2</sup></strong></span></span></div>
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<div class="row"><span class="left">Rearrange:</span><span class="right"><span class="intbl"><em>y<sup>2</sup></em><strong>x<sup>2</sup></strong></span> × <span class="intbl"><em>1</em><strong>2</strong></span>bc sin(A)</span></div>
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<div class="row"><span class="left">Which is:</span><span class="right"><span class="intbl"><em>y<sup>2</sup></em><strong>x<sup>2</sup></strong></span> × <b>Area of Triangle ABC</b></span></div>
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</div>
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<p>So we end up with this ratio:</p>
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<p align="center" class="large">Area of triangle ABC : Area of triangle PQR = x<sup>2 </sup>: y<sup>2</sup></p>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(1806, 1807, 3965, 1808, 1809, 3966, 1810, 1811, 3967, 3968);</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.html">Similar Triangles</a>
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<a href="triangles-similar-finding.html">Finding Similar Triangles</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="mean-proportional.html">Mean Proportional</a> <a href="../algebra/trigonometry-index.html">Trigonometry Index</a>
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