document.write( "Question 1085409: Find the perimeter of a triangle with sides 7 m, 8, and 10 m that has a uniform 1m border \n" ); document.write( "

Algebra.Com's Answer #699527 by ikleyn(52800) $\"\"$ $\"About$

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\n" ); document.write( "Obviously, it is an Olimpiad level problem.\r
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\n" ); document.write( "\n" ); document.write( "Usually, typical Olympiad level problem requires one non-trivial idea.\r
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\n" ); document.write( "\n" ); document.write( "This one requires TWO non-trivial ideas.\r
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\n" ); document.write( "\n" ); document.write( "These ideas are:\r
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document.write( "1.  The given triangle is SIMILAR to the second triangle formed by the uniform border.\r\n" );
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document.write( "    It is obvious: the sides are parallel, so the angles are congruent.\r\n" );
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document.write( "        * * * This is IDEA #1 * * * \r\n" );
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document.write( "2.  So, the only thing to discover is to find the proportionality (similarity) coefficient.\r\n" );
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document.write( "    Then we simply multiply the perimeter of the given triangle, 7 + 8 + 10 = 25 m, by the similarity coefficient.\r\n" );
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document.write( "3.  How to find the similarity coefficient ?   ??    ???   It is the question: TO BE OR NOT TO BE  ???\r\n" );
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document.write( "        Use the radius of the inscribed circle.  * * * It is the IDEA #2 * * * \r\n" );
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document.write( "    You can calculate the area of the given triangle using the Heron's formula.\r\n" );
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document.write( "    We all know this formula, so I will not bore with calculations and simply will give the answer: \r\n" );
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document.write( "        its area is A =  = 27.81 .\r\n" );
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document.write( "    Then the radius of the inscribed circle is  r =  =  = 2.225 (approximately; with 3 correct decimal digits after the decimal dot).\r\n" );
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document.write( "4.  Now only one step remains to the finish: The radius of the inscribed circle to the larger triangle is  r + 1 = 2.225 + 1 = 3.225 m.\r\n" );
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document.write( "    I don't know whether I should prove that the incentres of these triangles coincide: is is SO OBVIOUS . . . \r\n" );
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document.write( "5.  Now the similarity coefficient is  =  = 1.45 (larger to smaller) approximately with two correct decimal digits after the decimal dot.\r\n" );
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document.write( "Answer.  The perimeter under the question is 25*1.45 = 36.24 m.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "It is a nice problem. Thanks for submitting it. It was a pleasure to solve it.\r
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\n" ); document.write( "\n" ); document.write( "It is my reward after the daily fighting with idiotic, semi-idiotic, regular and routine problems . . . \r
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\n" ); document.write( "\n" ); document.write( "Thanks again.\r
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