document.write( "Question 1088506: please help me solve this problem \" Find the equation of the circle inscribed in a triangle, if the triangle has its sides on the lines ; x-y-3=0 , x+y-11=0 , and 7x+y-5=0. \n" ); document.write( "

Algebra.Com's Answer #702778 by ikleyn(52834) $\"\"$ $\"About$

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\n" ); document.write( "please help me solve this problem:
\n" ); document.write( "Find the equation of the circle inscribed in a triangle, if the triangle has its sides on the lines ; x-y-3=0 , x+y-11=0 , and 7x+y-5=0.
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document.write( "The equations of lines are\r\n" );
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document.write( " x - y =  3,    (1)\r\n" );
document.write( " x + y = 11,    (2)\r\n" );
document.write( "7x + y =  5.    (3)\r\n" );
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document.write( "1.  Notice that the lines (1) and (2) are perpendicular. So the triangle is right-angled.\r\n" );
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document.write( "2.  The intersection point of lines (1) and (2)  is  C = (7,4)  (you can find it mentally by adding equations (1) and (2) ).\r\n" );
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document.write( "    It is the right-angle vertex.\r\n" );
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document.write( "    The intersection point of lines (1) and (3)  is  A = (1,-2)  (you can find it mentally by adding equations (1) and (3) ).\r\n" );
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document.write( "    The intersection point of lines (2) and (3)  is  B = (-1,12)  (you can find it mentally by subtracting equations (2) from equation (3) ).\r\n" );
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document.write( "3.  The side CA is the vector (-6,-6) of the length .  It is the leg \"b\" of the triangle.\r\n" );
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document.write( "    The side CB is the vector (-8,8)  of the length . It is the leg \"a\" of the triangle.\r\n" );
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document.write( "    The side AB is the vector (-2,14) of the length .  It is the hypotenuse \"c\" of the triangle.\r\n" );
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document.write( "        (classic 3-4-5 right-angled triangle).\r\n" );
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document.write( "4.  It is well known fact that the radius of a right-angled triangle with the legs \"a\" and \"b\" and the hypotenuse \"c\" is equal to  ,\r\n" );
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document.write( "    which is in this case   = .\r\n" );
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document.write( "5.  The center of the circle lies on the angle bisector of the right angle at the vertex C.\r\n" );
document.write( "    This angle bisector is the horizontal line y = 4.\r\n" );
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document.write( "6.  The last thing to find is the x-coordinate of the center of the inscribed circle.\r\n" );
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document.write( "    This x-coordinate is equal to  7 -  = 7 -  = 7 -  = 7 - 4 = 3.\r\n" );
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document.write( "7.  Now we have everything to write the equation of the inscribed circle. It is \r\n" );
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document.write( "     = 8.    (8 =  =  ).\r\n" );
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document.write( "    The center is at (3,4); the radius is .\r\n" );
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