document.write( "Question 1035576: In a triangle ABC ,AB=12cms , BC=8cms, CA=10cms .An incircle is drawn inside the triangle
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Algebra.Com's Answer #650219 by ikleyn(52786) $\"\"$ $\"About$

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\n" ); document.write( "In a triangle ABC ,AB=12cms , BC=8cms, CA=10cms. An incircle is drawn inside the triangle touching the sides AB, BC and CA
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document.write( "I have found an appropriate figure from my archive  (see on the right).\r\n" );
document.write( "Although it is not in the scale, it is not so important for the solution.                 \r\n" );
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document.write( "The names a, b and c are just employed for the side lengths:\r\n" );
document.write( "|AB| = c = 12 cm, |BC| = a = 8 cm, and |AC| = b = 10 cm.\r\n" );
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document.write( "So we introduce d = |AD|= |AF|, e = |BE| = |BD| and f = |CF| = |CE|.\r\n" );
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document.write( "I hope you are familiar with the fact that tangent lines to a circle drawn \r\n" );
document.write( "from the outside point, are congruent, and therefore doubled equalities \r\n" );
document.write( "in the line above do not arose questions.\r\n" );
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document.write( "We have this system of equations\r\n" );
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document.write( "|AD| + |BD| = |AB|,                                 d + e = a,    (1)\r\n" );
document.write( "|BE| + |CE| = |BC|,       or, which is the same,    e + f = b.    (2)\r\n" );
document.write( "|AF| + |CF| = |AC|                                  d + f = c.    (3)\r\n" );
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document.write( "By adding equations (1), (2) and (3), you get\r\n" );
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document.write( "2(d + e + f) = a + b + c,   or    d + e + f = .     (4)\r\n" );
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Now distract equation (1) from (4).         You will get  f =  = .    (5)\r\n" );
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document.write( "Next, distract equation (2) from (4).       You will get  d =  = .     (6)\r\n" );
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document.write( "Similarly, distract equation (3) from (4).  You will get  e =  = .     (7)\r\n" );
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document.write( "Now substitute (plug in) the given numerical data into equations (5), (6) and (7). You will get\r\n" );
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document.write( "f =  = 7 cm,  d =  = 5 cm, e =  = 3 cm.\r\n" );
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document.write( "Answer.  AF = d = 5 cm,  BD = e = 3 cm, and CE = f = 7 cm.\r\n" );
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