document.write( "Question 1164223: the centres of three circles form a triangle PQR in which PQ=8cm, QR=10cm, and PR=12cm. If the circles are such that each touches the other two externally, find the radii of the circles \n" ); document.write( "

Algebra.Com's Answer #788535 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "Make a sketch, first.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Let \"a\", \"b\" and \"c\" be the circles radii, in ascending order.\r\n" );
document.write( "\r\n" );
document.write( "Then you have these equations\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    a + b =  8    (1)\r\n" );
document.write( "\r\n" );
document.write( "    a + c = 10    (2)\r\n" );
document.write( "\r\n" );
document.write( "    b + c = 12    (3)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "You should find three unknowns  \"a\", \"b\" and \"c\"  from these equations.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "For it, add the three equations (1), (2) and (3).  You will get\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    2a + 2b + 2c = 8 + 10 + 12 = 30,    \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "or, after dividing both sides by 2,\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    a + b + c = 15.     (4)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now subtract equation (1) from equation (4).  You will get\r\n" );
document.write( "\r\n" );
document.write( "    c = 15 - 8 = 7.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Next, subtract equation (2) from equation (4).  You will get\r\n" );
document.write( "\r\n" );
document.write( "    b = 15 - 10 = 5.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Finally, subtract equation (3) from equation (4).  You will get\r\n" );
document.write( "\r\n" );
document.write( "    a = 15 - 12 = 3.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The problem is just solved.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  The sides of the triangle are  7, 5 and 3 centimetres, in descending order.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "         // Fortunately, the triangle inequalities are satisfied with these numbers;\r\n" );
document.write( "            so such triangle does really exist (!)\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );