document.write( "Question 751500: Find the equation of the circle that is inscribed in the triangle whose sides lie on the lines 3x+y=5,x-3y=1,and x+3y=-7. \n" ); document.write( "
Algebra.Com's Answer #457347 by Edwin McCravy(20055)\"\" \"About 
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document.write( "Given three lines, there are 4 circles which are tangent to all\r\n" );
document.write( "three lines.  There are three excircles and one inscribed circle\r\n" );
document.write( "inside the triangle.  Here they are:\r\n" );
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document.write( "You only want the little red circle in the triangle, not the\r\n" );
document.write( "3 excircles.  However, we can't tell which of the four is the\r\n" );
document.write( "inscribed circle.  So we have to find them all.\r\n" );
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document.write( "Let the center of the circle be (h,k), and the radius r.  Then the\r\n" );
document.write( "equation of the circle is (x-h)² + (y-k)² = r²\r\n" );
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document.write( "We use the formula for perpendicular distance d \r\n" );
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document.write( "from the point whose coordinates are (x1,y1)\r\n" );
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document.write( "to the line whose equation is \r\n" );
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document.write( "Ax + By + C = 0 \r\n" );
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document.write( "That is given by this formula:\r\n" );
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document.write( "d = \"abs%28Ax%5B1%5D%2BBy%5B1%5D%2BC%29%2Fsqrt%28A%5E2%2BB%5E2%29\"\r\n" );
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document.write( "The perpendicular distances from the center to the three lines\r\n" );
document.write( "are all equal, since they are all three radii.\r\n" );
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document.write( "So we find all three distances from the center (h,k) to the \r\n" );
document.write( "three lines and we have:\r\n" );
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document.write( "We get 0 on the right side of the equations of the \r\n" );
document.write( "three lines\r\n" );
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document.write( "3x+y=5, x-3y=1, and x+3y=-7  \r\n" );
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document.write( "3x+y-5=0, x-3y-1=0, x+3y+7=0\r\n" );
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document.write( "Substituting in the formula above\r\n" );
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document.write( "r = \"abs%283h%2Bk-5%29%2Fsqrt%283%5E2%2B1%5E2%29\" = \"abs%28h-3k-1%29%2Fsqrt%281%5E2%2B%28-3%29%5E2%29\" = \"abs%28h%2B3k%2B7%29%2Fsqrt%281%5E2%2B3%5E2%29\"\r\n" );
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document.write( "r = \"abs%283h%2Bk-5%29%2Fsqrt%2810%29\" = \"abs%28h-3k-1%29%2Fsqrt%2810%29\" = \"abs%28h%2B3k%2B7%29%2Fsqrt%2810%29\"\r\n" );
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document.write( "So we have this system of equations:\r\n" );
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document.write( "\"abs%283h%2Bk-5%29%2Fsqrt%2810%29\" = \"abs%28h-3k-1%29%2Fsqrt%2810%29\"\r\n" );
document.write( "\"abs%28h-3k-1%29%2Fsqrt%2810%29\" = \"abs%28h%2B3k%2B7%29%2Fsqrt%2810%29\"\r\n" );
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document.write( "We multiply them both by √10 \r\n" );
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document.write( "|3h + k - 5| = |h - 3k - 1|\r\n" );
document.write( "|h - 3k - 1| = |h + 3k + 7|\r\n" );
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document.write( "There are four solutions to this system.  Here are the four systems\r\n" );
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document.write( "\"system%283h%2Bk-5=h-3k-1%2Ch-3k-1=h%2B3k%2B7%29\", \"system%283h%2Bk-5=h-3k-1%2Ch-3k-1=-%28h%2B3k%2B7%29%29\", \"system%283h%2Bk-5=-%28h-3k-1%29%2Ch-3k-1=h%2B3k%2B7%29%29\", \"system%283h%2Bk-5=-%28h-3k-1%29%2Ch-3k-1=-%28h%2B3k%2B7%29%29\"\r\n" );
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document.write( "Those simplify to\r\n" );
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document.write( "\"system%282h%2B4k=4%2C-6k=8%29\", \"system%282h%2B4k=4%2C2h=-6%29\",  \"system%284h-2k=6%2C-6k=8%29\",\"system%284h-2k=6%2C2h=-6%29\"\r\n" );
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document.write( "Their solutions are\r\n" );
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document.write( "(\"14%2F3\",-4/3), (-3,\"-5%2F2\"), (\"5%2F6\", \"-4%2F3\"), (-3,-9)\r\n" );
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document.write( "To find the radii, we substitute each of those in:\r\n" );
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document.write( "r = \"abs%283h%2Bk-5%29%2Fsqrt%2810%29\"\r\n" );
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document.write( "I'll just find the radius of the small red circle since that's\r\n" );
document.write( "the only one you were asked to find.\r\n" );
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document.write( "It is the circle with the center (\"5%2F6\", \"-4%2F3\").\r\n" );
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document.write( "r = \"abs%283%285%2F6%29%2B%28-4%2F3%29-5%29%29%2Fsqrt%2810%29\"\r\n" );
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document.write( "r = \"abs%285%2F2-4%2F3-5%29%2Fsqrt%2810%29\"\r\n" );
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document.write( "Multiply top and bottom by 6\r\n" );
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document.write( "r = \"abs%286%285%2F2-4%2F3-5%29%29%2F%286sqrt%2810%29%29\"\r\n" );
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document.write( "r = \"abs%2815-8-30%29%2F%286sqrt%2810%29%29\"\r\n" );
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document.write( "r = \"abs%28-23%29%2F%286sqrt%2810%29%29\"\r\n" );
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document.write( "r² = \"23%5E2%2F%2836%2A10%29\"\r\n" );
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document.write( "r² = \"529%2F360\"\r\n" );
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document.write( "Substituting in \r\n" );
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document.write( "(x-h)² + (y-k)² = r²\r\n" );
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document.write( "(x-\"5%2F6\")² + (y-(\"-4%2F3\"))² = \"529%2F360\"\r\n" );
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document.write( "(x-\"5%2F6\")² + (y+\"4%2F3\")² = \"529%2F360\"\r\n" );
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document.write( "Or you can multiply that out and simplify it in general form\r\n" );
document.write( "and get:\r\n" );
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document.write( "360x² + 360y² - 600x + 960y + 361 = 0\r\n" );
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document.write( "It doesn't come out very nice but it's correct.\r\n" );
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document.write( "Edwin
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