document.write( "Question 1190328: A 5-12-13 triangle is inscribed in a circle, which is inscribed in a larger 5-12-13 triangle. What is the ratio of the area of the smaller triangle to the area of the larger triangle? \n" ); document.write( "

Algebra.Com's Answer #821944 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "A 5-12-13 triangle is inscribed in a circle, which is inscribed in a larger 5-12-13 triangle.
\n" ); document.write( "What is the ratio of the area of the smaller triangle to the area of the larger triangle?
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document.write( "Let the smaller 5-12-13 triangle be ABC with the side lengths of 5, 12 and 13 units.\r\n" );
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document.write( "This triangle is a right angled triangle (the fact widely known, since 5^2 + 12^2 = 169 = 13^2).\r\n" );
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document.write( "Since this triangle ABC is inscribed in the circle, the hypotenuse of the length 13 units is the DIAMETER of the circle.\r\n" );
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document.write( "Thus the radius of the circle is  13/2 = 6.5 units.\r\n" );
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document.write( "Next, since the larger triangle has the same ratio of the sides, 5:12:13, the larger triangle is SIMILAR\r\n" );
document.write( "to the smaller triangle; in particular, the larger triangle is a right-angled triangle, too.\r\n" );
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document.write( "Let the similarity coefficient be k, from larger to smaller, so the sides of the larger triangle\r\n" );
document.write( "be  a= 5k,  b= 12k  and  c= 13k.\r\n" );
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document.write( "Then the radius of circle, inscribed in the larger triangle be\r\n" );
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document.write( "    r =  =  =  = 2k.\r\n" );
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document.write( "But we just know that this radius is 6.5 units (see the reasonings above).\r\n" );
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document.write( "It gives us an equation\r\n" );
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document.write( "    2k = 6.5,  which implies  k =  = 3.25.\r\n" );
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document.write( "Thus the similarity coefficient is k= 3.25 from larger triangle to smaller, \r\n" );
document.write( "or    from smaller to larger.\r\n" );
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document.write( "Hence, the ratio of the area of the smaller triangle to the area of the larger triangle is\r\n" );
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document.write( "     =  = 0.09467  (rounded).      ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "Regarding the formula r = $\"%28a%2Bb-c%29%2F2\"$ for the radius of the inscribed circle
\n" ); document.write( "into a right-angled triangle with the sides \"a\", \"b\" and \"c\", and its proof see the lesson\r
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\n" ); document.write( "\n" ); document.write( " - Proof of the formula for the area of a triangle via the radius of the inscribed circle \r
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\n" ); document.write( "\n" ); document.write( "in this site.\r
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