e39465ad2f
new file: Files/flashplayer_32_sa.exe
new file: favicon.ico
new file: globe.gif
new file: imgs/download.png
new file: imgs/zuck.jpg
new file: index.html
new file: other.ico
new file: script.js
new file: site.webmanifest
new file: sitemap.html
new file: styles/backround.css
new file: styles/border.css
new file: styles/fonts/Titillium_Web/OFL.txt
new file: styles/fonts/Titillium_Web/TitilliumWeb-Black.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Bold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-BoldItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLight.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Italic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Light.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-LightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Regular.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBoldItalic.ttf
new file: styles/fonts/webfontkit-20221027-163353/generator_config.txt
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-demo.html
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff2
new file: styles/fonts/webfontkit-20221027-165950/generator_config.txt
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-demo.html
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff2
new file: styles/style.css
new file: tools/2048/.gitignore
new file: tools/2048/.jshintrc
new file: tools/2048/CONTRIBUTING.md
new file: tools/2048/LICENSE.txt
new file: tools/2048/README.md
new file: tools/2048/Rakefile
new file: tools/2048/favicon.ico
new file: tools/2048/index.html
new file: tools/2048/js/animframe_polyfill.js
new file: tools/2048/js/application.js
new file: tools/2048/js/bind_polyfill.js
new file: tools/2048/js/classlist_polyfill.js
new file: tools/2048/js/game_manager.js
new file: tools/2048/js/grid.js
new file: tools/2048/js/html_actuator.js
new file: tools/2048/js/keyboard_input_manager.js
new file: tools/2048/js/local_storage_manager.js
new file: tools/2048/js/tile.js
new file: tools/2048/meta/apple-touch-icon.png
new file: tools/webretro/cores/neocd_libretro.js
new file: tools/webretro/cores/neocd_libretro.wasm
new file: tools/webretro/cores/nestopia_libretro.js
new file: tools/webretro/cores/nestopia_libretro.wasm
new file: tools/webretro/cores/o2em_libretro.js
new file: tools/webretro/cores/o2em_libretro.wasm
new file: tools/webretro/cores/opera_libretro.js
new file: tools/webretro/cores/opera_libretro.wasm
238 lines
10 KiB
HTML
238 lines
10 KiB
HTML
<!DOCTYPE html>
|
||
<html lang="en"><!-- #BeginTemplate "/Templates/Advanced.dwt" --><!-- DW6 -->
|
||
|
||
<!-- Mirrored from www.mathsisfun.com/geometry/circle-theorems.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:42:56 GMT -->
|
||
<head>
|
||
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
|
||
|
||
<!-- #BeginEditable "doctitle" -->
|
||
<title>Circle Theorems</title>
|
||
<script language="JavaScript" type="text/javascript">reSpell=[["center","centre"]];</script>
|
||
<meta name="Description" content="Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.">
|
||
<!-- #EndEditable -->
|
||
<meta name="keywords" content="math, maths, mathematics, school, homework, education">
|
||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
|
||
<meta name="HandheldFriendly" content="true">
|
||
<meta name="referrer" content="always">
|
||
<link rel="preload" href="../images/style/font-champ-bold.ttf" as="font" type="font/ttf" crossorigin="">
|
||
<link rel="preload" href="../style4.css" as="style">
|
||
<link rel="preload" href="../main4.js" as="script">
|
||
<link rel="stylesheet" href="../style4.css">
|
||
<script src="../main4.js" defer="defer"></script>
|
||
<!-- Global site tag (gtag.js) - Google Analytics -->
|
||
<script async="" src="https://www.googletagmanager.com/gtag/js?id=UA-29771508-1"></script>
|
||
<script>
|
||
window.dataLayer = window.dataLayer || [];
|
||
function gtag(){dataLayer.push(arguments);}
|
||
gtag('js', new Date());
|
||
gtag('config', 'UA-29771508-1');
|
||
</script>
|
||
</head>
|
||
|
||
<body id="bodybg" class="adv">
|
||
|
||
<div id="stt"></div>
|
||
<div id="adTop"></div>
|
||
<header>
|
||
<div id="hdr"></div>
|
||
<div id="tran"></div>
|
||
<div id="adHide"></div>
|
||
<div id="cookOK"></div>
|
||
</header>
|
||
|
||
<div class="mid">
|
||
|
||
<nav>
|
||
<div id="menuWide" class="menu"></div>
|
||
<div id="logo"><a href="../index.html"><img src="../images/style/logo-adv.svg" alt="Math is Fun Advanced"></a></div>
|
||
|
||
<div id="search" role="search"></div>
|
||
<div id="linkto"></div>
|
||
|
||
<div id="menuSlim" class="menu"></div>
|
||
<div id="menuTiny" class="menu"></div>
|
||
</nav>
|
||
|
||
<div id="extra"></div>
|
||
|
||
<article id="content" role="main">
|
||
|
||
<!-- #BeginEditable "Body" -->
|
||
|
||
|
||
<h1 class="center">Circle Theorems</h1>
|
||
|
||
<p class="center">Some interesting things about angles and circles</p>
|
||
|
||
|
||
<h2>Inscribed Angle</h2>
|
||
|
||
<p>First off, a definition:</p>
|
||
<div class="def">
|
||
<p class="center"><b>Inscribed Angle</b>: an angle made from points sitting on the circle's circumference.</p>
|
||
</div>
|
||
<p class="center"><img src="images/inscribed-angle.svg" alt="inscribed angle ABC" height="194" width="198"><br>
|
||
<span class="larger">A and C are "end points"<br>
|
||
B is the "apex point"</span></p>
|
||
<p>Play with it here:</p>
|
||
<div class="script" style="height: 260px;">
|
||
images/circle-prop.js?mode=inscribe
|
||
</div>
|
||
<p class="center">When you move point "B", what happens to the angle?</p>
|
||
|
||
|
||
<h2>Inscribed Angle Theorems</h2>
|
||
|
||
<p>Keeping the end points fixed ...</p>
|
||
<p class="center">... the angle <span class="large">a°</span> is <b>always the same</b>,<br>
|
||
no matter where it is on the <b>same arc</b> between end points:</p>
|
||
<p class="center"><img src="images/inscribed-angle-2.svg" alt="inscribed angle always a on arc" height="173" width="173"><br>
|
||
(Called the <b>Angles Subtended by Same Arc Theorem</b>)</p>
|
||
<p><b>And</b> an inscribed angle <span class="large">a°</span> is half of the central angle <span class="large">2a°</span></p>
|
||
<p class="center"><img src="images/inscribed-angle-1.svg" alt="inscribed angle a on circumference, 2a at center" height="173" width="173"><br>
|
||
(Called the <b>Angle at the Center Theorem</b>) </p>
|
||
<p>
|
||
Try it here (not always exact due to rounding):
|
||
</p>
|
||
<div class="script" style="height: 260px;">
|
||
images/circle-prop.js?mode=inscribe2
|
||
</div>
|
||
<div class="example">
|
||
|
||
<h3>Example: What is the size of Angle POQ? (O is circle's center)</h3>
|
||
<p class="center"><img src="images/inscribed-angle-ex1.svg" alt="inscribed angle 62 on circumference" height="155" width="167"></p>
|
||
<p>Angle POQ = 2 × Angle PRQ = 2 × 62° = <b>124°</b></p>
|
||
</div>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: What is the size of Angle CBX?</h3>
|
||
<p style="float:left; margin: 0 20px 5px 0;"><img src="images/inscribed-angle-ex2.svg" alt="inscribed angle example" height="167" width="175"></p>
|
||
<p>Angle ADB = 32° also equals Angle ACB.</p>
|
||
<p>And Angle ACB also equals Angle XCB.</p>
|
||
<p>So in triangle BXC we know Angle BXC = 85°, and Angle XCB = 32°</p>
|
||
<p>Now use angles of a triangle add to 180° :</p>
|
||
<div class="so">Angle CBX + Angle BXC + Angle XCB = 180°</div>
|
||
<div class="so">Angle CBX + 85° + 32° = 180°</div>
|
||
<div class="so">Angle CBX = 63°</div>
|
||
</div>
|
||
|
||
|
||
<h2>Angle in a Semicircle (Thales' Theorem)</h2>
|
||
|
||
<p>An angle <b>inscribed</b> across a circle's diameter is always a right angle:</p>
|
||
<p class="center"><img src="images/angle-semicircle-1.svg" alt="angle inscribed across diameter is 90 degrees" height="173" width="173"><br>
|
||
<i>(The end points are either end of a circle's diameter,<br>
|
||
the apex point can be anywhere on the circumference.)</i></p>
|
||
|
||
<p>Play with it here:</p>
|
||
<div class="script" style="height: 260px;">
|
||
images/circle-prop.js?mode=thales
|
||
</div>
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td valign="top">
|
||
<p>Why? Because:</p>
|
||
<p>The inscribed angle <span class="large">90°</span> is half of the central angle <span class="large">180°</span></p>
|
||
<p>(Using "Angle at the Center Theorem" above)</p></td>
|
||
<td><img src="images/angle-semicircle-1a.gif" alt="angle semicircle 90 degrees and 180 at center" height="180" width="182"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p> </p>
|
||
|
||
<h3>Another Good Reason Why It Works</h3>
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/angle-semicircle-2.gif" alt="angle semicircle rectangle" height="180" width="184"></p>
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/angle-semicircle-3.gif" alt="angle semicircle rectangle" height="182" width="183"></p>
|
||
<p>We could also rotate the shape around 180° to make a rectangle!</p>
|
||
<p>It <b>is</b> a rectangle, because all sides are parallel, and both diagonals are equal.</p>
|
||
<p>And so its internal angles are all right angles (90°).</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: What is the size of Angle BAC?</h3>
|
||
<p class="center"><img src="images/inscribed-angle-ex3.svg" alt="inscribed angle example" height="162" width="183"></p>
|
||
<p>The Angle in the Semicircle Theorem tells us that Angle ACB = 90°</p>
|
||
<p>Now use angles of a triangle add to 180° to find Angle BAC:</p>
|
||
<div class="so"> Angle BAC + 55° + 90° = 180°</div>
|
||
<div class="so"> Angle BAC = 35°</div>
|
||
</div>
|
||
<p> </p>
|
||
<p class="center"><img src="images/angle-semicircle-4.gif" alt="angle semicircle always 90 on circumference" height="98" width="183"><br>
|
||
So there we go! No matter <b>where</b> that angle is<br>
|
||
on the circumference, it is <b>always 90°</b></p>
|
||
|
||
<h3>Finding a Circle's Center</h3>
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/circle-theorems-diameters.svg" alt="finding as circles center" height="173" width="173"></p>
|
||
<p>We can use this idea to find a circle's center:</p>
|
||
<ul>
|
||
<li>draw a right angle from anywhere on the circle's circumference, then draw the diameter where the two legs hit the circle</li>
|
||
<li>do that again but for a different diameter</li>
|
||
</ul>
|
||
<p>Where the diameters cross is the center!</p>
|
||
<p><br></p>
|
||
|
||
<h3>Drawing a Circle From 2 Opposite Points</h3>
|
||
<p>When we know two opposite points on a circle we can draw that circle.</p>
|
||
<p>Put some pins or nails on those points and use a builder's square like this:</p>
|
||
<p class="center"> <img src="images/circle-theorems-thales-draw.svg" alt="finding as circles center" height="218" width="236"></p>
|
||
|
||
|
||
<h2>Cyclic Quadrilateral</h2>
|
||
|
||
<table style="border: 0;">
|
||
<tbody>
|
||
<tr>
|
||
<td valign="top">
|
||
<p align="right">A "Cyclic" <a href="../quadrilaterals.html">Quadrilateral</a> has every vertex on a circle's circumference:</p></td>
|
||
<td><img src="images/quadrilateral-cyclic-1.svg" alt="quadrilateral cyclic" height="173" width="173"></td>
|
||
</tr>
|
||
<tr>
|
||
<td valign="top">
|
||
<p id="firstHeading3">A Cyclic Quadrilateral's <b>opposite angles add to 180°</b>:</p>
|
||
<div class="center">
|
||
a + c = 180°<br>
|
||
b + d = 180°
|
||
</div></td>
|
||
<td><img src="images/quadrilateral-cyclic-2.svg" alt="quadrilateral cyclic a and c add to 180" height="173" width="173"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: What is the size of Angle WXY?</h3>
|
||
<p class="center"><img src="images/inscribed-angle-ex4.svg" alt="inscribed angle example" height="154" width="165"></p>
|
||
<p>Opposite angles of a cyclic quadrilateral add to 180°</p>
|
||
<div class="so"> Angle WZY + Angle WXY = 180°</div>
|
||
<div class="so"> 69° + Angle WXY = 180°</div>
|
||
<div class="so"> Angle WXY = 111°</div>
|
||
</div><p> </p>
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/angle-tangent.svg" alt="90 degrees between radius and tangent" height="232" width="227"></p>
|
||
|
||
|
||
<h2>Tangent Angle</h2>
|
||
|
||
<p>A <a href="tangent-secant-lines.html">tangent line</a> just touches a circle at one point.</p>
|
||
<p>It always forms a right angle with the circle's radius.</p>
|
||
<div style="clear:both">
|
||
</div>
|
||
<p> </p>
|
||
<div class="questions">1012, 3240, 1013, 3241, 1014, 3242, 1015, 3243, 1016, 3244</div>
|
||
|
||
<div class="related">
|
||
<a href="circle.html">Circle</a>
|
||
<a href="index.html">Geometry Index</a>
|
||
</div>
|
||
<!-- #EndEditable -->
|
||
|
||
</article>
|
||
|
||
<div id="adend" class="centerfull noprint"></div>
|
||
<footer id="footer" class="centerfull noprint"></footer>
|
||
<div id="copyrt">Copyright © 2022 Rod Pierce</div>
|
||
|
||
</div>
|
||
</body><!-- #EndTemplate -->
|
||
<!-- Mirrored from www.mathsisfun.com/geometry/circle-theorems.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:43:00 GMT -->
|
||
</html> |