document.write( "Question 85404: I really need help with this one, can someone please help me\r
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Algebra.Com's Answer #61596 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
I really need help with this one, can someone please help me\r
\n" ); document.write( "\n" ); document.write( "Use coordinate geometry to prove that the quadrilateral OPQR
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document.write( "We have to show that OP II QR, i.e., that both\r\n" );
document.write( "have the same slope. \r\n" );
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document.write( "mOP = \"%28+%283%29-%280%29%29%2F%28%289%29-%280%29%29\" = \"3%2F9\" = \"1%2F3\"\r\n" );
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document.write( "mQR = \"%28+%284%29-%288%29%29%2F%28%28-3%29-%289%29%29\" = \"%28-4%29%2F%28-12%29\" = \"1%2F3\"\r\n" );
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document.write( "Their slopes are equal, so OPQR is either a trapezoid or a parallelogram.\r\n" );
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document.write( "Now let's show that PQ = RO\r\n" );
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document.write( "DPQ = \"sqrt%28%289-9%29%5E2+%2B+%288-3%29%5E2%29\" = \"sqrt%28%280%29%5E2+%2B+%285%29%5E2%29\" =\"sqrt%280+%2B+25%29\" = \"sqrt%2825%29\" = \"5\" \r\n" );
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document.write( "DRO = \"sqrt%28%280-%28-3%29%29%5E2+%2B+%280-4%29%5E2%29\" = \"sqrt%28%283%29%5E2+%2B+%28-4%29%5E2%29\" = \"sqrt%289+%2B+16%29\" = \"sqrt%2825%29\" = \"5\"\r\n" );
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document.write( "You might think this is enough to show that that OPQR is an isosceles\r\n" );
document.write( "trapezoid since one pair of opposite sides are parallel and the other \r\n" );
document.write( "pair of opposite sides are equal in length.\r\n" );
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document.write( "However, that could be said about a rectangle.  We must rule out\r\n" );
document.write( "a rectangle. (Yes we can look and see that it's not a rectangle,\r\n" );
document.write( "but \"looking and seeing\" is not acceptable. The easiest way to do \r\n" );
document.write( "this is to rule out a right angle using slopes. We will show that\r\n" );
document.write( "OP is not perpendicular to RO\r\n" );
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document.write( "We have already found the slope of OP as 1/3. So we will rule out\r\n" );
document.write( "OPQR as being a rectangle by showing that the slope of RO is not\r\n" );
document.write( "the \"negative reciprocal\" of 1/3.  That is to show that the slope of\r\n" );
document.write( "RO is not -3.\r\n" );
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document.write( "mRO = \"%28+%284%29-%280%29%29%2F%28%28-3%29-%280%29%29\" = \"4%2F%28-3%29\" = \"-4%2F3\"\r\n" );
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document.write( "\"-4%2F3\" is not equal to \"-3\" so we have ruled out a rectangle.\r\n" );
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document.write( "So OPQR must be an isosceles trapezoid.\r\n" );
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document.write( "Edwin
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