document.write( "Question 1163372: Two bicycles are driving on the circle in the same direction with speeds of 9 mph and 5 mph respectively. How many points are there on the circle where the two bicycles meet( and it doesn’t have a radius but there is a correct answer) \n" ); document.write( "

Algebra.Com's Answer #787453 by ikleyn(52852) $\"\"$ $\"About$

You can put this solution on YOUR website!
.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " I will assume that they started from the same point SIMULTANEOUSLY \r
\n" ); document.write( "\n" ); document.write( " (although the post missed this important condition).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Let  L  be the circumference of the circle.\r\n" );
document.write( "\r\n" );
document.write( "Then the faster cyclist will catch the slower cyclist first time  when the faster cyclist will cover \r\n" );
document.write( "the distance which exactly 1 circumference longer than the distance covered by the slower cyclist \r\n" );
document.write( "\r\n" );
document.write( "      9t - 5t = L.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It gives the time to get first meeting point\r\n" );
document.write( "\r\n" );
document.write( "       t =  =   hours\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "and the distance which the faster cyclist covered during this time is\r\n" );
document.write( "\r\n" );
document.write( "        = 9t =   miles.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The distance which the slower cyclist covered during this time is \r\n" );
document.write( "\r\n" );
document.write( "        = 5t = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The meeting point is geometrically the same point on the circle for both cyclists, , \r\n" );
document.write( "and its angle measure on the circle is\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "         =  =  =   of the full angle of  radians, or 90 degrees.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, they started simultaneously, and their first meeting point is at the 90 degrees angle.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Next, they started from this point  SIMULTANEOUSLY  and . . . and everything was repeated.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Hence, their next meeting point is the point on the circle with the angle of 180 degrees.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    It can be proved using strict mathematical arguments, but it should be ABSOLUTELY CLEAR to you without any arguments.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, there are 4 remarkable points on the circle: first point is the starting point, and 3 other points \r\n" );
document.write( "\r\n" );
document.write( "(the points where whey meet/catch each other) are the images of the starting point, rotated 90°, 180°, and 270° along the circle.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  There are 4 points on the circle, where the faster cyclist catches the slower cyclist.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Pretty amazing and unexpected result (!)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );