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document.write( "I went rummaging through the garbage of the last deluge of bad AI \"solutions\"\r\n" );
document.write( "and found this interesting rhombus problem buried there. (incorrectly done by\r\n" );
document.write( "AI.)\r\n" );
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document.write( "Rhombuses are easier to think about when you draw them diamond-shaped, i.e.,\r\n" );
document.write( "symmetrical with the horizontal and the vertical. So I will draw the figure\r\n" );
document.write( "that way instead of the way it's drawn on the site of the given link. \r\n" );
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document.write( "As you can see from the second figure below, any rhombus can be partitioned into\r\n" );
document.write( "8 CONGRUENT isosceles triangles (they might look equilateral but that's not\r\n" );
document.write( "necessarily the case. They are only isosceles, I just accidentally drew them to\r\n" );
document.write( "look equilateral.)\r\n" );
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document.write( "Anyway, the second figure below shows that ΔTPW is 1/8 of the rhombus \r\n" );
document.write( "(area-wise). That's too obvious to bother wasting time to prove. \r\n" );
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document.write( "Quadrilateral PWVT is made up of ΔTPW and ΔTVW. So all that's left is to find\r\n" );
document.write( "what fraction ΔTVW is of the whole rhombus and add that to 1/8.\r\n" );
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document.write( "We will use these two well-known theorems:\r\n" );
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document.write( "Theorem 1: If two triangles are similar, the ratio of their areas is equal to\r\n" );
document.write( "the square of the ratio of any pair of corresponding sides between them. \r\n" );
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document.write( "Theorem 2: The intersection of the three medians of a triangle, called the\r\n" );
document.write( "centroid, is located two-thirds of the distance from each vertex to the midpoint\r\n" );
document.write( "of the opposite side. That is to say, the shorter part of each median from the\r\n" );
document.write( "centroid is 1/2 the longer part from the centroid. It also says that the\r\n" );
document.write( "entire median is 3 times the shorter part when divided by the centroid.\r\n" );
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document.write( "Notice that SW, QT and PO are the three medians of ΔSPQ, and V is the centroid\r\n" );
document.write( "of ΔSPQ.\r\n" );
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document.write( "ΔSVQ is 1/3 of ΔSPQ. Why? Remembering the formula for the area of a triangle,\r\n" );
document.write( "A=1/2(bh), they have the same base SQ, and by theorem 2, the height OV of ΔSVQ\r\n" );
document.write( "is 1/3 of the height OP of ΔSPQ. \r\n" );
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document.write( "Since ΔSVQ is 1/3 of ΔSPQ which is 1/2 of the whole rhombus, then ΔSVQ is\r\n" );
document.write( "(1/3)(1/2) = 1/6 of the whole rhombus. \r\n" );
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document.write( "By theorem 2, VW is 1/2 of SV, and since ΔTVW is similar to ΔSVQ, the area\r\n" );
document.write( "of ΔTVW, by theorem 1, is (1/2)2=1/4 the area of ΔSVQ\r\n" );
document.write( "So ΔTVW is (1/4)(1/6) = 1/24th of the whole rhombus.\r\n" );
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document.write( "Therefore quadrilateral PWVT is 1/8 + 1/24 = 3/24 + 1/24 = 4/24 = 1/6 of rhombus PQRS.\r\n" );
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document.write( "Answer: 1/6\r\n" );
document.write( " \r\n" );
document.write( "Edwin
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