document.write( "Question 397657: Two points are chosen at random on a line segment whose length is a > 0. Find the probability that the 3 line segments thus formed can be the sides of a triangle. \n" ); document.write( "

Algebra.Com's Answer #281797 by Edwin McCravy(20086) $\"\"$ $\"About$

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document.write( "The other tutor is right! Here's a bit more geometrical drawings:  \r\n" );
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document.write( "Suppose this is the line segment.\r\n" );
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document.write( "0------------x----------y------------a\r\n" );
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document.write( "Then the sides of the triangle are x, y-x, and a-y\r\n" );
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document.write( "The sum of any two sides must be greater than the third side.\r\n" );
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document.write( "So\r\n" );
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document.write( "x + (y-x) > a-y,    x + (a-y) > y-x,     (y-x) + (a-y) > x      y > x\r\n" );
document.write( "x + y - x > a-y     x + a - y > y-x            y-x+a-y > x \r\n" );
document.write( "        y > a-y       2x - 2y > -a                -x+a > x   \r\n" );
document.write( "       2y > a        -2x + 2y < a                    a > 2x\r\n" );
document.write( "        y > a/2       2y - 2x < a                  a/2 > x   \r\n" );
document.write( "                       2(y-x) < a                    x < a/2   \r\n" );
document.write( "                          y-x <           \r\n" );
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document.write( "We also know that y < a  and x > 0\r\n" );
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document.write( "So we have a system of inequalities:\r\n" );
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document.write( "We draw the boundary lines:\r\n" );
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document.write( "and find that the solution to the system of inequalities is the triangle\r\n" );
document.write( "marked \"SOLUTION\".  It's area is \r\n" );
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document.write( "So the numerator of the probability is the area of that triangle, for\r\n" );
document.write( "the coordinates of any point within that triangle will represent points\r\n" );
document.write( "on the original line:\r\n" );
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document.write( "0------------x----------y------------a\r\n" );
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document.write( "that satisy the system of inequalities.\r\n" );
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document.write( "The denominator of the probability is the area of this triangle:\r\n" );
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document.write( "Because the coordinates of any point inside this triangle represents \r\n" );
document.write( "a pair of values for x and y that could have been picked on this line:\r\n" );
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document.write( "0------------x----------y------------a\r\n" );
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document.write( "The area of this triangle is \r\n" );
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document.write( "So the desired probability is  \r\n" );
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document.write( "Edwin

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