document.write( "Question 251045: Points P,Q, R are 3 vertices of a parallelogram. Find the coordinates of point D, the 4th vertex. Find 3 solutions. What are the 3 solutions ? \n" ); document.write( "
Algebra.Com's Answer #182822 by Edwin McCravy(20056) $\"\"$ $\"About$
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Points P,Q, R are 3 vertices of a parallelogram. Find the coordinates of point D, the 4th vertex. Find 3 solutions. What are the 3 solutions ?
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document.write( "You forgot to give the coordinates or P, Q and R, so I'll\r\n" );
document.write( "just make some up and you can model your solution after this\r\n" );
document.write( "one using the correct coordinates:\r\n" );
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document.write( "I'll suppose that P is the point P(2,3), Q is the point Q(8,1),\r\n" );
document.write( "and Q is the point R is R(11,5), \r\n" );
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document.write( "Those three points form a triangle. \r\n" );
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document.write( "To make a parallelogram out of a triangle, you make a triangle\r\n" );
document.write( "congruent to it, reverse it, then paste it on one of the three\r\n" );
document.write( "sides of the triangle.  Since a triangle has three sides,\r\n" );
document.write( "there are three sides you can paste it on.\r\n" );
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document.write( "You can make one parallelogram like this by pasting it on the \r\n" );
document.write( "right side:\r\n" );
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document.write( "In that case we let the 4th point be S(x,y),  Then for\r\n" );
document.write( "RS and PQ to be parallel,\r\n" );
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document.write( "1.  the slope of the line RS must be equal to the slope \r\n" );
document.write( "of PQ, so we use the slope formula to set those slopes \r\n" );
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document.write( "Reducing the fraction on the right:\r\n" );
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document.write( "Cross-multiplying,\r\n" );
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document.write( "and also for SQ to be parallel to RP,\r\n" );
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document.write( "1.  the slope of the line SQ must be equal to the slope \r\n" );
document.write( "of RP, so we use the slope formula to set those slopes \r\n" );
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document.write( "So we solve the system:\r\n" );
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document.write( "and we get S(17,3)\r\n" );
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document.write( "Or you can make a parallelogram by pasting it on the\r\n" );
document.write( "top side, like this:\r\n" );
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document.write( "In that case we let the 4th point be T(x,y),  Then for\r\n" );
document.write( "RT and QP to be parallel,\r\n" );
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document.write( "1.  the slope of the line RT must be equal to the slope \r\n" );
document.write( "of QP, so we use the slope formula to set those slopes \r\n" );
document.write( "equal:\r\n" );
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document.write( "Reducing the fraction on the right:\r\n" );
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document.write( "Cross-multiplying,\r\n" );
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document.write( "So we solve the system:\r\n" );
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document.write( "and we get T(5,7)\r\n" );
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document.write( "Or you can make a parallelogram by pasting on the\r\n" );
document.write( "bottom side like this:\r\n" );
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document.write( "In that case we let the 4th point be U(x,y),  Then for\r\n" );
document.write( "PU and RQ to be parallel,\r\n" );
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document.write( "1.  the slope of the line PU must be equal to the slope \r\n" );
document.write( "of RQ, so we use the slope formula to set those slopes \r\n" );
document.write( "equal:\r\n" );
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document.write( "Simplifying the fraction on the right:\r\n" );
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document.write( "Cross-multiplying,\r\n" );
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document.write( "and also for QU to be parallel to RP,\r\n" );
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document.write( "2.  the slope of the line QU must be equal to the slope \r\n" );
document.write( "of RP, so we use the slope formula to set those slopes \r\n" );
document.write( "equal:\r\n" );
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document.write( "Simplifying the fraction on the right:\r\n" );
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document.write( "So we solve the system:\r\n" );
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document.write( "and we get U(-1,-1)\r\n" );
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document.write( "Now so you can see how they all look togethsr, you\r\n" );
document.write( "have one big triangle, similar to the original triangle,\r\n" );
document.write( "and of course to all the parts, and with the sides twice\r\n" );
document.write( "as long as the original's sides.\r\n" );
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document.write( "Edwin
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