document.write( "Question 766795: four points P,Q,R,S are such that PQ=10,QR=30,RS=15 and PS=m. If m is an integer and no three of these points lie on a straight line, what is the number of possible values of m? \n" ); document.write( "

Algebra.Com's Answer #467232 by Edwin McCravy(20060) $\"\"$ $\"About$

You can put this solution on YOUR website!
four points P,Q,R,S are such that PQ=10,QR=30,RS=15 and PS=m. If m is an integer and no three of these points lie on a straight line, what is the number of possible values of m?
\n" ); document.write( "

 \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Notice in the drawing that QR = QB+BC+CR = 10+5+15 = 30 units, as given.\r\n" );
document.write( "\r\n" );
document.write( "Let's first of all assume that it would not matter if three of the\r\n" );
document.write( "points P,Q,R,S could lie on a straight line.  Then  \r\n" );
document.write( "\r\n" );
document.write( "P could be any point on the circle on the left.\r\n" );
document.write( "S could be any point on the circle on the right.\r\n" );
document.write( "\r\n" );
document.write( "Therefore PS is longest when P is at A and S is at D.\r\n" );
document.write( "In that case the length of PS would be \r\n" );
document.write( "\r\n" );
document.write( "AD = AQ+QB+BC+CR+RD = 10+10+5+15+15 = 55 units\r\n" );
document.write( "\r\n" );
document.write( "and PS is shortest when P is at B and S is at C.\r\n" );
document.write( "In that case the length of PS would be \r\n" );
document.write( "\r\n" );
document.write( "BC = 5 units\r\n" );
document.write( "\r\n" );
document.write( "Since m is an integer it could be any of these values\r\n" );
document.write( "\r\n" );
document.write( "5,6,7,...,53,54,55\r\n" );
document.write( "\r\n" );
document.write( "But when we require that no three of P,Q,R,S can lie on a\r\n" );
document.write( "straight line, we must rule out 5 and 55, so the possible\r\n" );
document.write( "values are \r\n" );
document.write( "\r\n" );
document.write( "6,7,8,...,52,53,54\r\n" );
document.write( "\r\n" );
document.write( "There are 54 integers from 1 through 54 inclusive, and \r\n" );
document.write( "we cannot use 1,2,3,4 or 5, so there are 54-5 = 49 \r\n" );
document.write( "possible integer values for m, the length of PS.\r\n" );
document.write( "\r\n" );
document.write( "Answer: 49\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );