document.write( "Question 1060920: In the given figure, line DB bisects the exterior angle EBA of triangle ABC. If AB = 6, BC = 10, and AC = 12, find DA.\r
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Algebra.Com's Answer #675843 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "In the given figure, line DB bisects the exterior angle EBA of triangle ABC. If AB = 6, BC = 10, and AC = 12, find DA.\r
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document.write( "1.  Apply the Cosine Law and find cosine of the angle ABC:\r\n" );
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document.write( "     = ,  or\r\n" );
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document.write( "     = ,  which gives\r\n" );
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document.write( "    cos(ABC) =  =  = .\r\n" );
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document.write( "    In particular, the angle ABC is acute and lies in QI.\r\n" );
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document.write( "2.  The angle ABE is the supplement angle to the angle ABC.\r\n" );
document.write( "    Therefore, the angle ABE is obtuse, lies in QII and cos(ABE) = -cos(ABC) = .\r\n" );
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document.write( "3.  The angle ABD is the half of the angle ABE.\r\n" );
document.write( "    Find cosine of the angle ABD by applying the formula of cosine of the half angle:\r\n" );
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document.write( "    cos(ABD) =  =  =  = .\r\n" );
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document.write( "4.  Let \"x\" be the length of the segment AD abd \"y\" be the length of the segment BD:  x = |AD|,  y = |BD|.\r\n" );
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document.write( "    Apply the Cosine Law to the triangle ABD:\r\n" );
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document.write( "     = .           (1)\r\n" );
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document.write( "    Apply the Cosine Law to the triangle CBD:\r\n" );
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document.write( "     = .   (2)\r\n" );
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document.write( "5.  Now solve the system of two nonlinear equations in two unknowns (1) and (2).\r\n" );
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document.write( "    For it, open parentheses in (2) and distract eq.(1) from eq.(2). You will get\r\n" );
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document.write( "    24x + 144 = ,  or\r\n" );
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document.write( "    24x = .            (3)\r\n" );
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document.write( "Now you have one linear equation (3) with the expressed \"x\" via \"y\", and the quadratic equation (1).\r\n" );
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document.write( "It is solvable. You can solve it and obtain expressions for \"x\" and \"y\".\r\n" );
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document.write( "In this way you will complete the solution and will get the answer.\r\n" );
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