document.write( "Question 1155099: The sides of triangle XYZ are XY = XZ = 25 and YZ = 40. A semicircle is inscribed in triangle XYZ so that its diameter lies on YZ, and is tangent to the other two sides. Find the area of the semicircle. \n" ); document.write( "

Algebra.Com's Answer #777693 by ikleyn(52786) $\"\"$ $\"About$

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document.write( "You are given the isosceles triangle XYZ.\r\n" );
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document.write( "Draw the perpendicular (the height, the altitude) from the vertex X to the base YZ.\r\n" );
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document.write( "Let A be the foot of this altitude.\r\n" );
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document.write( "According to the WELL KNOWN property of isosceles triangles, the altitude XA is the median at the same time.\r\n" );
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document.write( "Thus the original triangle XYZ is divided in two congruent right angled triangles XYA and XZA.\r\n" );
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document.write( "Their legs YA and ZA are 40/2 = 20 units long (each).\r\n" );
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document.write( "So, each of these two right angled triangles is (3,4,5) triangles with the hypotenuse of 25 inits long and \r\n" );
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document.write( "with the legs of 20 units long (YA and ZA)  and 15 units long (the altitude XA).\r\n" );
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document.write( "The area of each of the two right angled triangles is   = 150 square units.\r\n" );
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document.write( "Now I will calculate the area of the triangle XYA by another way.\r\n" );
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document.write( "Draw the altitude in the triangle XYA from its vertex A to its hypotenuse XY.\r\n" );
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document.write( "Notice that this altitude's length is equal to the radius of the inscribed semi-circle (!)\r\n" );
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document.write( "  +---------------------------------------------+\r\n" );
document.write( "  |                                             |\r\n" );
document.write( "  |  It is the major point of the solution (!)  |\r\n" );
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document.write( "So, if \"r\" be the radius of the seni-circle, then for the area of the triangle XYA we have this equation\r\n" );
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document.write( "     = 150,    or\r\n" );
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document.write( "     25*r     = 300.\r\n" );
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document.write( "Hence,  r =  = 12 units.\r\n" );
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document.write( "So, the problem is just solved, and the  ANSWER  is  12 units for the radius and  \r\n" );
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document.write( " =  square units for the area of the semi-circle.\r\n" );
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