document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199312>Question 1199312</A>: <I><SPAN STYLE=\"word-break: break-all;\"> let O(0,0), A(6,0), B(6,6), c(0,6) be the vertices of a square OABC, and Let M be the midpoint of OB. Find the probability that a point chosen at random from the square is <BR>\n" );
document.write( "a) father from O than from M<BR>\n" );
document.write( "b) more than twice as far from O as from M </SPAN>\n" );
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document.write( "let O(0,0), A(6,0), B(6,6), c(0,6) be the vertices of a square OABC, and Let M be the midpoint of OB.<br><BR>\n" );
document.write( "Here is a sketch:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-1%2C7%2C-1%2C7%0D%0A%2Cline%280%2C0%2C6%2C0%29%2Cline%286%2C0%2C6%2C6%29%2Cline%286%2C6%2C0%2C6%29%2Cline%280%2C6%2C0%2C0%29%2Cline%280%2C0%2C6%2C6%29%0D%0A%2Clocate%280%2C-.2%2C%22O%280%2C0%29%22%29%2Clocate%286%2C-.2%2C%22A%286%2C0%29%22%29%2Clocate%286%2C6.5%2C%22B%286%2C6%29%22%29%2Clocate%280%2C6.5%2C%22C%280%2C6%29%22%29%0D%0A%2Ccircle%283%2C3%2C.1%29%2Clocate%283.2%2C3.2%2C%22M%283%2C3%29%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "a) Find the probability that a point chosen at random from the square is farther from O than from M.<br><BR>\n" );
document.write( "To do this, determine where the points are that are the same distance from O as they are from M.  This is simple; they lie on the perpendicular bisector of segment OM.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-1%2C7%2C-1%2C7%0D%0A%2Cline%280%2C0%2C6%2C0%29%2Cline%286%2C0%2C6%2C6%29%2Cline%286%2C6%2C0%2C6%29%2Cline%280%2C6%2C0%2C0%29%2Cline%280%2C0%2C6%2C6%29%0D%0A%2Clocate%280%2C-.2%2C%22O%280%2C0%29%22%29%2Clocate%286%2C-.2%2C%22A%286%2C0%29%22%29%2Clocate%286%2C6.5%2C%22B%286%2C6%29%22%29%2Clocate%280%2C6.5%2C%22C%280%2C6%29%22%29%0D%0A%2Ccircle%283%2C3%2C.1%29%2Clocate%283.2%2C3.2%2C%22M%283%2C3%29%22%29%0D%0A%2Cline%283%2C0%2C0%2C3%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Note that this result can be obtained algebraically by solving the equation that says the distance from P(x,y) to O(0,0) is the same as the distance from P(x,y) to M(3,3):<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E2%2By%5E2%29=sqrt%28%28x-3%29%5E2%2B%28y-3%29%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%5E2%2By%5E2%29=sqrt%28%28x-3%29%5E2%2B%28y-3%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(I won't do that in my response; however, it would be a good exercise for the student to solve this equation and find that indeed the set of points equidistant from O and M are the points on the line x+y=3.)<br><BR>\n" );
document.write( "It should be easy to see from the sketch that the points in the square that are farther from O than from M constitute exactly 7/8 of the square.  So<br><BR>\n" );
document.write( "ANSWER to part a): 7/8<br><BR>\n" );
document.write( "b) Find the probability that a point chosen at random from the square is more than twice as far from O as from M.<br><BR>\n" );
document.write( "To solve this part, we could find the points that are EXACTLY twice as far from O as from M by solving the equation<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E2%2By%5E2%29=2%2A%28sqrt%28%28x-3%29%5E2%2B%28y-3%29%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%5E2%2By%5E2%29=2%2A%28sqrt%28%28x-3%29%5E2%2B%28y-3%29%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(Again I won't do that here in my response; and again it would be a good exercise for the student to do so.  You should find that the circle is centered at (4,4) with radius 2*sqrt(2).)<br><BR>\n" );
document.write( "The set of points exactly twice as far from O as from M is a circle.  Two of those points are (2,2) and (6,6); and by symmetry the segment joining those two points is a diameter of the circle.<br><BR>\n" );
document.write( "Here is a sketch:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-1%2C7%2C-1%2C7%0D%0A%2Cline%280%2C0%2C6%2C0%29%2Cline%286%2C0%2C6%2C6%29%2Cline%286%2C6%2C0%2C6%29%2Cline%280%2C6%2C0%2C0%29%2Cline%280%2C0%2C6%2C6%29%0D%0A%2Clocate%280%2C-.2%2C%22O%280%2C0%29%22%29%2Clocate%286%2C-.2%2C%22A%286%2C0%29%22%29%2Clocate%286%2C6.5%2C%22B%286%2C6%29%22%29%2Clocate%280%2C6.5%2C%22C%280%2C6%29%22%29%0D%0A%2Ccircle%283%2C3%2C.1%29%2Clocate%283.2%2C3.2%2C%22M%283%2C3%29%22%29%0D%0A%2Ccircle%282%2C2%2C.1%29%2Clocate%282.2%2C2.2%2C%22%282%2C2%29%22%29%2Ccircle%284%2C4%2C.1%29%2Clocate%284.2%2C4.2%2C%22%284%2C4%29%22%29%0D%0A%2Ccircle%284%2C4%2Csqrt%288%29%29%0D%0A%2Ccircle%286%2C2%2C.1%29%2Clocate%286%2C2.2%2C%22%286%2C2%29%22%29%2Ccircle%282%2C6%2C.1%29%2Clocate%282%2C6%2C%22%282%2C6%29%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To answer this part of the problem, we need to find what fraction of the square is inside the circle.<br><BR>\n" );
document.write( "The region of the square inside the circle consists of a semicircle plus an isosceles right triangle determined by the segment from (6,2) to (2,6).  The radius of the circle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%282%29%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the lengths of the legs of the triangle are 4, so the area of the semicircle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%28pi%29%28%282%2Asqrt%282%29%29%5E2%29=4pi ALIGN=MIDDLE  ALT=\"%281%2F2%29%28pi%29%28%282%2Asqrt%282%29%29%5E2%29=4pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the area of the triangle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%284%29%284%29=8 ALIGN=MIDDLE  ALT=\"%281%2F2%29%284%29%284%29=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "The area of the whole square is 36, so the probability that a random point in the square is more than twice as far from O as it is from M is<br><BR>\n" );
document.write( "ANSWER to part b): <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284pi%2B8%29%2F36=%28pi%2B2%29%2F9 ALIGN=MIDDLE  ALT=\"%284pi%2B8%29%2F36=%28pi%2B2%29%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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