document.write( "Question 193994: The lengths of the longest and shortest sides of an acute scalene triangle are 9 m nd 41 m .what could be the length of third side? \n" ); document.write( "
Algebra.Com's Answer #145681 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
the lengths of the longest and shortest sides of an acute
\n" ); document.write( "scalene triangle are 9 m and 41 m, what could be the length
\n" ); document.write( "of third side?
\n" ); document.write( "
\r\n" );
document.write( "Let a = 9m be the one side of the triangle\r\n" );
document.write( "Let b = 41m be the second side of the triangle\r\n" );
document.write( "Let c = the unknown third side.\r\n" );
document.write( "\r\n" );
document.write( "Since c is the middle-size side, and b=41m is the\r\n" );
document.write( "longest side, we know that c < 41m\r\n" );
document.write( "\r\n" );
document.write( "Let the angle between sides a and c be angle B, which is also\r\n" );
document.write( "the angle opposite side b.\r\n" );
document.write( "By the law of cosines:\r\n" );
document.write( "b2 = a2 + c2 - 2ac*cos(B)\r\n" );
document.write( "\r\n" );
document.write( "Solve for cos(B)\r\n" );
document.write( "\r\n" );
document.write( "              a2 + c2 - b2\r\n" );
document.write( "    cos(B) =  ------------\r\n" );
document.write( "                  2ac\r\n" );
document.write( "              92 + c2 - 412 \r\n" );
document.write( "    cos(B) =  ------------\r\n" );
document.write( "                 2(9)c\r\n" );
document.write( "\r\n" );
document.write( "              81 + c2 - 1681 \r\n" );
document.write( "    cos(B) =  ------------\r\n" );
document.write( "                 2(9)c\r\n" );
document.write( "\r\n" );
document.write( "              c2 - 1600 \r\n" );
document.write( "    cos(B) =  ------------\r\n" );
document.write( "                 2(9)c\r\n" );
document.write( "\r\n" );
document.write( "              c2 - 1600 \r\n" );
document.write( "    cos(B) =  ------------\r\n" );
document.write( "                  18c\r\n" );
document.write( "\r\n" );
document.write( "In order for B to be less than 90°, cos(B) must be greater than 0.\r\n" );
document.write( "\r\n" );
document.write( "So            cos(B) > 0\r\n" );
document.write( "\r\n" );
document.write( "or\r\n" );
document.write( "\r\n" );
document.write( "       c2 - 1600 \r\n" );
document.write( "       ------------  > 0\r\n" );
document.write( "            18c\r\n" );
document.write( "\r\n" );
document.write( "Multiplying through by 18c will not reverse\r\n" );
document.write( "the inequality since c is positive.\r\n" );
document.write( "\r\n" );
document.write( "       c2 - 1600 > 0       \r\n" );
document.write( "\r\n" );
document.write( "Factor the left side:\r\n" );
document.write( "\r\n" );
document.write( "    (c - 40)(c + 40) > 0   \r\n" );
document.write( "\r\n" );
document.write( "This is only true when c > 40\r\n" );
document.write( "\r\n" );
document.write( "So the answer is that c must be between 40m and 41m,\r\n" );
document.write( "exclusive of both 40m and 41m. That can be written:\r\n" );
document.write( "\r\n" );
document.write( "   40 < c < 41\r\n" );
document.write( "\r\n" );
document.write( "Edwin

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