document.write( "Question 374262: Express the length L of a chord of a circle with radius 10cm as a function of the central angle \n" ); document.write( "
Algebra.Com's Answer #266359 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "There are two methods.  The answers look different but are\r\n" );
document.write( "equivalent.  I'll do it both ways:\r\n" );
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document.write( "If you have studied the law of cosines then:\r\n" );
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document.write( "\"L%5E2=10%5E2%2B10%5E2-2%2810%29%2810%29cos%28theta%29\"\r\n" );
document.write( "\"L%5E2=100%2B100-200cos%28theta%29\"\r\n" );
document.write( "\"L%5E2=200-200cos%28theta%29\"\r\n" );
document.write( "\"L%5E2=200%281-cos%28theta%29%29\"\r\n" );
document.write( "\"L+=+sqrt%28+200%281-cos%28theta%29%29+%29\"\r\n" );
document.write( "\"L+=+sqrt%28100%2A2%281-cos%28theta%29%29+%29\"\r\n" );
document.write( "\"L+=+10sqrt%282%281-cos%28theta%29%29%29\"\r\n" );
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document.write( "If you haven't studied the law of cosines, it can be done another way,\r\n" );
document.write( "but it involves the angle \"theta%2F2\" instead of \"theta\".\r\n" );
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document.write( "For doing it that way, draw in this green median, which, since the triangle\r\n" );
document.write( "is isoceles, is the bisector of the central angle as well as the \r\n" );
document.write( "perpendicular bisector of the chord, forming two congruent right triangles.\r\n" );
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document.write( "Looking at the upper right triangle only, we see that\r\n" );
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document.write( "\"sin%28theta%2F2%29\"\"%22%22=%22%22\"\"%28L%2F2%29%2F10\"\r\n" );
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document.write( "Multiply both sides by 10\r\n" );
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document.write( "\"10sin%28theta%2F2%29\"\"%22%22=%22%22\"\"L%2F2\"\r\n" );
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document.write( "Multiply both sides by 2\r\n" );
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document.write( "\"20sin%28theta%2F2%29\"\"%22%22=%22%22\"\"L\"\r\n" );
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document.write( "Edwin
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