document.write( "Question 363490: The parallel sides of trapezoid ABCD are 3 cm and 9 cm(AB and DC).The non parallel sides are 4 cm and 6 cm(AD and BC).A line(PQ) parallel to base(DC) divides the trapezoid into two trapezoids(ABQP and PQCD ) of equal perimeters. Find the ratio in which each of the non parallel sides is divided,if AP:BQ=PD:QC \n" ); document.write( "
Algebra.Com's Answer #259717 by Edwin McCravy(20056)\"\" \"About 
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document.write( "Note: the word is \"perimeter\", not \"parameteras\", as you had it.\r\n" );
document.write( "\"Perimeter\" means the distance all the way around a figure.\r\n" );
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document.write( "I corrected that :-)\r\n" );
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\n" ); document.write( "The parallel sides of trapezoid ABCD are 3 cm and 9 cm(AB and DC).The non parallel sides are 4 cm and 6 cm(AD and BC).A line(PQ) parallel to base(DC) divides the trapezoid into two trapezoids(ABQP and PQCD ) of equal perimeters. Find the ratio in which each of the non parallel sides is divided,if AP:BQ=PD:QC
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document.write( "Here's the trapezoid:\r\n" );
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document.write( "Now we'll draw in PQ parallel to the top and bottom of the trapezoid,\r\n" );
document.write( "cutting it into two trapezoids with equal perimeters. \r\n" );
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document.write( "Now I'll explain my labeling of those parts:\r\n" );
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document.write( "\"AP%2FBQ+=+PD%2FQC\"\r\n" );
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document.write( "\"AP%2AQC+=+PD%2ABQ\"\r\n" );
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document.write( "Divide both sides by \"PD%2AQC\"\r\n" );
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document.write( "\"%28AP%2AQC%29%2F%28PD%2AQC%29+=+%28PD%2AQC%29%2F%28PD%2AQC%29\"\r\n" );
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document.write( "\"AP%2FPD+=+BQ%2FQC\"\r\n" );
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document.write( "Take reciprocals of both sides\r\n" );
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document.write( "\"PD%2FAP+=+QC%2FBQ\"\r\n" );
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document.write( "Add 1 to both sides\r\n" );
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document.write( "\"PD%2FAP+%2B+1=+QC%2FBQ%2B1\"\r\n" );
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document.write( "Replace 1 by \"AP%2FAP\" on the left and replace 1 on the right by \"BQ%2FBQ\"\r\n" );
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document.write( "\"PD%2FAP+%2B+AP%2FAP=+QC%2FBQ+%2B+BQ%2FBQ\"\r\n" );
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document.write( "Combine numerators over denominators:\r\n" );
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document.write( "\"%28PD%2BAP%29%2FAP+=+%28BQ%2BQC%29%2FBQ\"\r\n" );
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document.write( "Since AP+PD = AD = 6 and BQ+QC = BC = 4\r\n" );
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document.write( "\"6%2FAP+=+4%2FBQ\"\r\n" );
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document.write( "Take reciprocals of both sides\r\n" );
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document.write( "\"AP%2F6+=+BQ%2F4\"\r\n" );
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document.write( "Let that ratio be = k\r\n" );
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document.write( "\"AP%2F6+=+BQ%2F4+=+k\"\r\n" );
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document.write( "\"AP+=+6k\" and \"BQ=4k\"\r\n" );
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document.write( "BQ = 4k,\r\n" );
document.write( "QC = 4 - 4k\r\n" );
document.write( "AP = 6k\r\n" );
document.write( "PD = 6 - 6k \r\n" );
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document.write( "all in centimeters. \r\n" );
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document.write( "Perimeter of the upper trapezoid ABQP:\r\n" );
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document.write( "BQ + AB + AP + PQ\r\n" );
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document.write( "Perimeter of the lower trapezoid PQCD:\r\n" );
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document.write( "QC + DC + PD + PQ\r\n" );
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document.write( "Equating the two perimeters:\r\n" );
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document.write( "BQ + QB + AP + PQ = QC + DC + PD + PQ\r\n" );
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document.write( "Subtract PQ (their common side) from both sides:\r\n" );
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document.write( "     BQ + QB + AP = QC + DC + PD\r\n" );
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document.write( "Substitute their lengths in terms of the fraction k:\r\n" );
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document.write( "      4k + 3 + 6k = (4-4k) + 9 + (6-6k)\r\n" );
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document.write( "          10k + 3 = 4 - 4k + 9 + 6 - 6k\r\n" );
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document.write( "          10k + 3 = 19 - 10k\r\n" );
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document.write( "              20k = 16\r\n" );
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document.write( "                k = \"16%2F20\" = \"4%2F5\" = 0.8\r\n" );
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document.write( "Therefore\r\n" );
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document.write( "AP = 6k = 6(0.8) = 4.8 cm.  \r\n" );
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document.write( "BQ = 4k = 4(0.8) = 3.2 cm. \r\n" );
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document.write( "PD = 6-6k = 6-6(0.8) = 1.2 cm.\r\n" );
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document.write( "QC = 4-4k = 4-4(0.8) = 0.8 cm\r\n" );
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document.write( "We want to find\r\n" );
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document.write( "AP:PD = BQ:QC\r\n" );
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document.write( "which is 4.8:1.2 = 48:12 = 4:1\r\n" );
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document.write( "and checking:\r\n" );
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document.write( "BQ:QC = 3.2:0.8 = 32:8 = 4:1\r\n" );
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document.write( "That's the answer 4:1\r\n" );
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document.write( "Incidentally, the figures above are drawn approximately to scale.\r\n" );
document.write( "Measure them with a ruler and you will see they are fairly close to\r\n" );
document.write( "3cm, 4cm, 9cm and 6cm.\r\n" );
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document.write( "The perimeters are equal even though the upper trapezoid is obviously \r\n" );
document.write( "lots bigger in area.  This demonstrates the important fact that two\r\n" );
document.write( "figures can have the same perimeter and quite different areas.  \r\n" );
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document.write( "Edwin

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