document.write( "Question 675193: Write a coordinate proof for each statement. The three segments joining the midpoints of the sides of an isosceles triangle form another isosceles triangle. \n" ); document.write( "
Algebra.Com's Answer #419789 by Edwin McCravy(20060)\"\" \"About 
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document.write( "If the black triangle below is isosceles, and the vertices of the green\r\n" );
document.write( "triangle are the midpoints of the sides of the black triangle, that the\r\n" );
document.write( "green triangle is also isosceled:\r\n" );
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document.write( "Since the black triangle is isosceles we can let it be triangle PQR,\r\n" );
document.write( "where the vertices are P(0,b), Q(-a,0), and R(a,0), where a and b are\r\n" );
document.write( "both positive numbers:\r\n" );
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document.write( "Now we will let S, T. and U be the midpoints respectively of\r\n" );
document.write( "PQ, PR and QR.\r\n" );
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document.write( "We will use the midpoint formula to find the coordinates of S,T, and U:\r\n" );
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document.write( "For S:\r\n" );
document.write( "Midpoint = \r\n" );
document.write( "Midpoint of PQ = S\"%28matrix%281%2C3%2C++++++%28-a%2B0%29%2F2%2C+++%22%2C%22%2C+%280%2Bb%29%2F2%29%29\"\r\n" );
document.write( "Midpoint of PQ = S\"%28matrix%281%2C3%2C++++++-a%2F2%2C+++%22%2C%22%2C+b%2F2%29%29\"\r\n" );
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document.write( "For T:\r\n" );
document.write( "Midpoint = \r\n" );
document.write( "Midpoint of PR = T\"%28matrix%281%2C3%2C++++++%280%2Ba%29%2F2%2C+++%22%2C%22%2C+%28b%2B0%29%2F2%29%29\"\r\n" );
document.write( "Midpoint of PR = T\"%28matrix%281%2C3%2C++++++a%2F2%2C+++%22%2C%22%2C+b%2F2%29%29\"\r\n" );
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document.write( "For U:\r\n" );
document.write( "Midpoint = \r\n" );
document.write( "Midpoint of QR = U\"%28matrix%281%2C3%2C++++++%28-a%2Ba%29%2F2%2C+++%22%2C%22%2C+%280%2B0%29%2F2%29%29\"\r\n" );
document.write( "Midpoint of QR = U(0,0)   \r\n" );
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document.write( "Now all we need is to show that US = UT.\r\n" );
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document.write( "We will use the distance tance formula to show that:\r\n" );
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document.write( " d = \"sqrt%28%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2%29\"\r\n" );
document.write( "SU = \"sqrt%28%280-%28-a%2F2%29%29%5E2%2B%280-b%2F2%29%5E2%29\"\r\n" );
document.write( "SU = \"sqrt%28%28a%2F2%29%5E2%2B%28-b%2F2%29%5E2%29\"\r\n" );
document.write( "SU = \"sqrt%28a%5E2%2F2%5E2%2Bb%5E2%2F2%5E2%29\"\r\n" );
document.write( "SU = \"sqrt%28a%5E2%2F4%2Bb%5E2%2F4%29\"\r\n" );
document.write( "SU = \"sqrt%28%28a%5E2%2Bb%5E2%29%2F4%29\"\r\n" );
document.write( "SU = \"sqrt%28a%5E2%2Bb%5E2%29%2F2\"\r\n" );
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document.write( " d = \"sqrt%28%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2%29\"\r\n" );
document.write( "TU = \"sqrt%28%28a%2F2-0%29%5E2%2B%28b%2F2-0%29%5E2%29\"\r\n" );
document.write( "TU = \"sqrt%28%28a%2F2%29%5E2%2B%28b%2F2%29%5E2%29\"\r\n" );
document.write( "TU = \"sqrt%28a%5E2%2F2%5E2%2Bb%5E2%2F2%5E2%29\"\r\n" );
document.write( "TU = \"sqrt%28a%5E2%2F4%2Bb%5E2%2F4%29\"\r\n" );
document.write( "TU = \"sqrt%28%28a%5E2%2Bb%5E2%29%2F4%29\"\r\n" );
document.write( "TU = \"sqrt%28a%5E2%2Bb%5E2%29%2F2\"\r\n" );
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document.write( "So SU = TU and therefore triangle STU is isosceles.\r\n" );
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document.write( "Edwin
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