document.write( "Question 1143730: Determine the equation of the circle inscribed in a triangle, if the triangle has its sides on the lines y=0, 4x+3y-50=0, and 3x-4y=0. Sketch the graph. \n" ); document.write( "
Algebra.Com's Answer #764637 by ikleyn(52788)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The lines are\r\n" );
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document.write( "          y      = 0     (1)\r\n" );
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document.write( "    4x + 3y - 50 = 0     (2)\r\n" );
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document.write( "    3x - 4y      = 0     (3)\r\n" );
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document.write( "    \r\n" );
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document.write( "    Lines  y=0 (red), 4x+3y-50 = 0 (green), 3x-4y = 0 (blue)\r\n" );
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document.write( "The triangle is in QI. Its vertices are\r\n" );
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document.write( "    P1 = (0,0)        (the intersection points of lines (1) and (3) )\r\n" );
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document.write( "    P2 = (12.5,0)     (the intersection point of lines  (1) and (2) )\r\n" );
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document.write( "    P3 = (8,6)        (the intersection point of lines  (2) and (3) )\r\n" );
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document.write( "Let the point (p,q) be the center of the inscribed circle into the triangle.\r\n" );
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document.write( "        Then the point (p,q) is EQUALLY REMOTED from the triangle sides.\r\n" );
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document.write( "So, I will write two equations. One equation will say that the point (p,q) is equally remoted from lines (1) and (2).\r\n" );
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document.write( "Seqond equation will say that the point (p,q) is equally remoted from lines (1) and (3).\r\n" );
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document.write( "The formula for the distance of the point (p,q) on a coordinate plane from the line given by its general equation Ax + By - C = 0 is\r\n" );
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document.write( "    d = \"abs%28Ap%2BBq+%2B+C%29%2Fsqrt%28A%5E2%2BB%5E2%29\".        (4)\r\n" );
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document.write( "So, my first equation is\r\n" );
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document.write( "    \"abs%280p%2B1q+%2B+0%29%2Fsqrt%280%5E2%2B1%5E2%29\" = \"abs%284p%2B3q+-+50%29%2Fsqrt%284%5E2%2B3%5E2%29\",\r\n" );
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document.write( "or\r\n" );
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document.write( "    \"abs%28q%29%2F1\" = \"abs%284p%2B3q+-+50%29%2F5\".      (5)\r\n" );
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document.write( "The center of the triangle lies, obviously, in QI  --  therefore, I can take off the modulus sign in the left part at \"q\"\r\n" );
document.write( "   (since q is positive in QI (!) )\r\n" );
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document.write( "The center of the triangle lies, obviously, BELOW line (2) -- therefore, 4p+3q-50 < 0 (!).  Therefore, I can take off\r\n" );
document.write( "the modulus sign in the right part, replacing it by the sign \" - \" (minus).\r\n" );
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document.write( "Then equation (5) takes the form\r\n" );
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document.write( "    q = -(4p + 3q -50)/5,  \r\n" );
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document.write( "or, in standard equivalent form\r\n" );
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document.write( "    4p + 8q = 50.     (6)\r\n" );
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document.write( "My second equation is\r\n" );
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document.write( "    \"abs%280p%2B1q+%2B+0%29%2Fsqrt%280%5E2%2B1%5E2%29\" = \"abs%283p-4q%29%2Fsqrt%284%5E2%2B3%5E2%29\",\r\n" );
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document.write( "or\r\n" );
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document.write( "    \"abs%28q%29%2F1\" = \"abs%283p-4q%29%2F5\".      (7)\r\n" );
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document.write( "Again, similarly to what was said above, I can take off the modulus sign in the left part at \"q\"\r\n" );
document.write( "   (since q is positive in QI (!) )\r\n" );
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document.write( "The center of the triangle lies, obviously, BELOW line (3) -- therefore, 3p-4q > 0 (!).  Therefore, I can take off\r\n" );
document.write( "the modulus sign in the right part, replacing it by the sign \" + \" (plus).\r\n" );
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document.write( "Then equation (7) takes the form\r\n" );
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document.write( "    q = (3p - 4q)/5,  \r\n" );
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document.write( "or, in standard equivalent form\r\n" );
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document.write( "    3p - 9q = 0.      (8)\r\n" );
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document.write( "Now I have two equations (6) and (8) to find p and q. \r\n" );
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document.write( "    4p + 8q = 50,     (6)\r\n" );
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document.write( "    3p - 9q =  0.     (8)\r\n" );
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document.write( "Simplify them\r\n" );
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document.write( "    2p + 4q = 25      (6')\r\n" );
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document.write( "     p - 3q =  0      (8')\r\n" );
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document.write( "From (8'), express p = 3q and substitute it into (6'). You will get\r\n" );
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document.write( "    2*(3q) + 4q = 25,\r\n" );
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document.write( "    6q     + 4q = 25\r\n" );
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document.write( "    10q      = 25\r\n" );
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document.write( "      q = \"25%2F10\" = 2.5.\r\n" );
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document.write( "Then p = 3q = 7.5.\r\n" );
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document.write( "The radius of the circle r is, obviously, equal to q:  r = q = 2.5.\r\n" );
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document.write( "Therefore, the equation of the circle is\r\n" );
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document.write( "    \"%28x-p%29%5E2\" + \"%28y-q%29%5E2\" = \"r%5E2\",   or\r\n" );
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document.write( "    \"%28x-7.5%29%5E2\" + \"%28y-2.5%29%5E2\" = \"2.5%5E2\".     ANSWER\r\n" );
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document.write( "You can transform it further as you want.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Regarding the basic equation (4),  see the lessons\r
\n" ); document.write( "\n" ); document.write( "    - The distance from a point to a straight line in a coordinate plane \r
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\n" ); document.write( "\n" ); document.write( "    - Using formula for the distance from a point to a straight line in a plane to solve word problems \r
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