document.write( "Question 1103133: The perimeter of triangle ABC is 24. M is the midpoint of AB such that MC=MA=5. How many square inches are in the area of the triangle? \n" ); document.write( "

Algebra.Com's Answer #717810 by ikleyn(53765) $\"\"$ $\"About$

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document.write( "The key intermediate STATEMENT is that this triangle ABC is a right-angled triangle.\r\n" );
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document.write( "One can prove it by different ways.  I choose this one:\r\n" );
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document.write( "    The fact that the point M is equidistant from the points A, B and C means that the point M is the center of the circle \r\n" );
document.write( "    subscribed about the triangle ABC.\r\n" );
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document.write( "    AB is the diameter of this circle.\r\n" );
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document.write( "    Hence, the angle C is the right angle, since it leans on the diameter.\r\n" );
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document.write( "Thus we have the right angled triangle ABC.\r\n" );
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document.write( "Its hypotenuse AB has the length 5 + 5 = 10 units     (given !).\r\n" );
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document.write( "The sum of the legs is the perimeter minus hypotenuse = 24 - 10 = 14 units.\r\n" );
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document.write( "Thus we have this system of two equations for the legs x and y:\r\n" );
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document.write( "x + y = 14,          (1)\r\n" );
document.write( "x^2 + y^2 = 10^2.    (2)\r\n" );
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document.write( "From (1) express y = 14-x and substitute it into (2).\r\n" );
document.write( "You will get the quadratic equation for x:\r\n" );
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document.write( "x^2 + (14-x)^2 = 100,\r\n" );
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document.write( "x^2 + 196 - 28x + x^2 = 100,\r\n" );
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document.write( "2x^2 - 28x + 96 = 0,\r\n" );
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document.write( "x^2 - 14x + 48 = 0,\r\n" );
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document.write( "(x-6)*(x-8) = 0.\r\n" );
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document.write( "Thus we have this pair of solutions (x,y) = (6,8)  or  this pair  (x,y) = (8,6).\r\n" );
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document.write( "In any case, the area of the triangle ABC is  = 24 square units.\r\n" );
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