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105 lines
4.9 KiB
HTML
105 lines
4.9 KiB
HTML
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<h1 class="center">Intersecting Chords Theorem</h1>
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<p>This is the idea (a,b,c and d are lengths):</p>
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<p class="center"><img src="images/circle-intersect-chords.svg" alt="Intersecting chords" ></p>
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<p>And here it is with some actual values (measured only to whole numbers):</p>
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<p class="center"><img src="images/circle-intersect-chords-ex.svg" alt="chords example" ></p>
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<p>And we get</p>
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<ul>
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<li>71 × 104 = <b>7384</b></li>
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<li>50 × 148 = <b>7400</b></li>
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</ul>
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<p>Very close! If we measured perfectly the results would be <b>equal</b>.</p>
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<p> </p>
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<p class="center larger">Why not try drawing one yourself, measure the lengths and see what you get?</p>
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<p> </p>
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<h3>Why is this true?</h3>
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<p>Because there are similar triangles! Looking below:</p>
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<li>The angles <span class="larger">θ</span> are the same (see <a href="vertically-opposite-angles.html">vertically opposite angles</a>)</li>
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<li>The angles <span class="larger">φ</span> are the same (see <a href="circle-theorems.html">inscribed angles</a>)</li>
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</ul>
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<p class="center"><img src="images/circle-intersect-chords-reason.svg" alt="Intersecting chords reason" ></p>
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<p>The triangles may not be the same size, but they have the same angles ... so all lengths will be in proportion! </p>
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<p>One triangle has the ratio a/c, and the other has the matching ratio d/b:</p>
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<p class="so">a/c = d/b</p>
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<p class="so">a × b = c × d</p>
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<a href="circle.html">Circle</a> <a href="tangent-secant-lines.html"> <a href="tangent-secant-lines.html">Tangent and Secant Lines</a> <a href="index.html">Geometry Index</a>
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