document.write( "Question 388720: A point S is chosen inside the square MNPQ. What is the probability that the angle MSN is a right angle? Express your answer as a decimal to the nearest hundredths?\r
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Algebra.Com's Answer #275089 by Edwin McCravy(20060) $\"\"$ $\"About$

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A point S is chosen inside the square MNPQ. What is the probability that the angle MSN is a right angle? Express your answer as a decimal to the nearest hundredths?
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document.write( "The other tutor's answer to the second part is incorrect.\r\n" );
document.write( "We draw the square:\r\n" );
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document.write( "The angle MSN will be a right angle if and only if the arc it subtends\r\n" );
document.write( "has measure 180°, that is, the arc is a semicircle.\r\n" );
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document.write( "So we draw a circle with diameter MN:\r\n" );
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document.write( "S must be picked along the right side of the green circle, say like this:\r\n" );
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document.write( "in order for angle MSN to be a right angle.\r\n" );
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document.write( "So S must be picked as one of the points of a 1 dimensional semicircle out of\r\n" );
document.write( "one of the points of a 2 dimensional area.  But a semicircle is just a curved\r\n" );
document.write( "line and curved (or straight) lines have 0 thickness and thus 0 area.\r\n" );
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document.write( "So the answer is 0.\r\n" );
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document.write( "A point S is chosen inside the square MNPQ.  What is the probability that the\r\n" );
document.write( "angle MSN is acute? Express your answer as a decimal to the nearest hundredths?\r\n" );
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document.write( "Now this problem is not 0. For if point S chosen in this part of the area:\r\n" );
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document.write( "like this:\r\n" );
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document.write( "then angle MSN will necessarily be an acute angle.\r\n" );
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document.write( "So the desired probability is the area of this\r\n" );
document.write( "figure:\r\n" );
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document.write( "divided by the area of the whole square\r\n" );
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document.write( "Suppose \" r \"  represents the radius of the green circle:\r\n" );
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document.write( "Then each side of the square MNPQ has length 2r, so its area is (2r)² or 4r².\r\n" );
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document.write( "The area of the green circle is pr²\r\n" );
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document.write( "So the semicircle that is cut away from the square in this figure\r\n" );
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document.write( "is one-half of that area or \r\n" );
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document.write( "So the area of that figure is the area of the square minus the\r\n" );
document.write( "area of the semicircle or \r\n" );
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document.write( "That's the numerator of the desired probability.  The denominator is the\r\n" );
document.write( "area of the square MNPQ or 4r², so the desired probability is\r\n" );
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document.write( " =\r\n" );
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document.write( " =\r\n" );
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document.write( " =\r\n" );
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document.write( " = 0.6073009183\r\n" );
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document.write( "Rounded to tenths the probability is 0.6\r\n" );
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document.write( "So the probability that angle MSN is acute is about 0.6,\r\n" );
document.write( "the probability that it is obtuse is about 0.4, and\r\n" );
document.write( "the probability that it is a right angle is 0.  Seems\r\n" );
document.write( "strange that that should be 0, since it is possible that\r\n" );
document.write( "it could be a right angle, but the probability of an angle\r\n" );
document.write( "being exactly 90° out of an infinite number of possibilities\r\n" );
document.write( "is loss than 1 out of a trillion trillion! which could only be\r\n" );
document.write( "0.\r\n" );
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document.write( "Edwin

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