document.write( "Question 1210173: Let P be a point chosen uniformly at random inside triangle ABC. Extend ray BP to hit side AC at D. What is the probability that BD < 4?\r
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\n" ); document.write( "\n" ); document.write( "The sides of triangle ABC are 3, 5, and 7. \n" ); document.write( "
Algebra.Com's Answer #851472 by ikleyn(52803)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "Let P be a point chosen uniformly at random inside triangle ABC. Extend ray BP to hit side AC at D.
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\n" ); document.write( "\n" ); document.write( "        After checking my previous solution one more time, I found an error in it.\r
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\n" ); document.write( "\n" ); document.write( "        After that, I fixed the error, and now I place this my updated solution here AGAIN.\r
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\n" ); document.write( "\n" ); document.write( "        Edwin was right in that my previous solution was incorrect.\r
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\n" ); document.write( "\n" ); document.write( "        Sorry for that and thanks to Edwin for helping me to detect my error.\r
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\n" ); document.write( "\n" ); document.write( "        Below you see this my corrected version.\r
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\n" ); document.write( "\n" ); document.write( "                            S O L U T I O N\r
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document.write( "I will borrow the picture from the post by Edwin with my \"thanks\".\r\n" );
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document.write( "    \r\n" );
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document.write( "From the given side lengths, we \r\n" );
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document.write( "    (a)  will determine cos(A) using the cosine low;\r\n" );
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document.write( "    (b)  will determine a position of the point D' in the figure;\r\n" );
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document.write( "    (c)  then the probability is the ratio of the area of triangle BAD' to the area of triangle BAC,\r\n" );
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document.write( "         and this ratio is nothing else as the ratio of the length AD' to AC.       .\r\n" );
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document.write( "Below is the implementation of these steps.\r\n" );
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document.write( "(a)  According to the cosine law, \r\n" );
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document.write( "        7^2 = 3^2 + 5^2 - 2*3*5*cos(A).\r\n" );
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document.write( "     It gives\r\n" );
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document.write( "        cos(A) = \"%283%5E2%2B5%5E2-7%5E2%29%2F%282%2A3%29\" = -0.5.\r\n" );
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document.write( "(b)  We want to find a point D' such that the length of BD' is 4.\r\n" );
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document.write( "     Let BD' be x.\r\n" );
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document.write( "     Use the cosine law again\r\n" );
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document.write( "        4^2 = 3^2 + x^2 - 2*3*x*cos(A),\r\n" );
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document.write( "        4^2 = 3^2 + x^2 - 2*3*x*(-1/2),\r\n" );
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document.write( "        4^2 = 3^2 + x^2 + 3x\r\n" );
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document.write( "        x^2 + 3x + 3^2 - 4^2 = 0,\r\n" );
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document.write( "        x^2 + 3x - 7 = 0\r\n" );
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document.write( "        \"x%5B1%2C2%5D\" = \"%28-3+%2B-+sqrt%283%5E2+%2B+4%2A7%29%29%2F2\" = \"%28-3+%2B-+sqrt%2837%29%29%2F2\".\r\n" );
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document.write( "     Reject the negative root and accept the positive one  x = \"%28sqrt%2837%29-3%29%2F2\" = 1.541381265.\r\n" );
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document.write( "(c)  Thus the length of BD' is 1.541381265 unit.\r\n" );
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document.write( "     So, the probability under the problem's question is  the ratio of AD' to AC, \r\n" );
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document.write( "     which is  \"1.541381265%2F5\" = 0.308276253.\r\n" );
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document.write( "     Notice that numerically it is the same answer as in the post by Edwin.\r\n" );
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document.write( "ANSWER.  The probability under the problem's question is  \"%28sqrt%2837%29-3%29%2F10\", or  0.308276253,  approximately.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Addition.\r
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\n" ); document.write( "\n" ); document.write( "This addition is for Edwin and for those who tracked the events around this problem.\r
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\n" ); document.write( "\n" ); document.write( "Edwin criticized me for the statement \r
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document.write( "    (c)  then the probability is the ratio of the area of triangle BAD' to the area of triangle BAC,\r\n" );
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document.write( "         and this ratio is nothing else as the ratio of the length AD' to AC.   \r\n" );
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\n" ); document.write( "\n" ); document.write( "This my statement is  \"highlight%28highlight%28absolutely%29%29\"  \"highlight%28highlight%28correct%29%29\",  and it  \"highlight%28highlight%28STAYS%29%29\"  \"highlight%28highlight%28FIRMLY%29%29\"  as a  \"highlight%28highlight%28ROCK%29%29\".\r
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\n" ); document.write( "\n" ); document.write( "The error was in deriving the quadratic equation.
\n" ); document.write( "Originally,  I derived it with the error,  but when I fixed this error,  I got everything correct.\r
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\n" ); document.write( "\n" ); document.write( "Edwin asks me\r
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document.write( "    How do you know this:\r\n" );
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document.write( "        The ratio of the area of triangle BAD' to the area of triangle BAC \r\n" );
document.write( "        equals the ratio of the length AD' to AC.\r\n" );
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document.write( "    What theorem shows this is true? \r\n" );
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\n" ); document.write( "\n" ); document.write( "Edwin, I will prove it right now. The proof is a simple Geometry exercise.\r
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document.write( "Prolong the line CA from 'C' to 'A' and further to the left and up.\r\n" );
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document.write( "Draw a perpendicular from vertex B to this prolonged line CA to the intersection at some point E.\r\n" );
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document.write( "Then BE is the common height/altitude for triangles BAD' and BAC;\r\n" );
document.write( "the bases for these triangles are AD' and AC, respectively.  \r\n" );
document.write( "Let 'h' denotes the length of BE:  h = |BE|.\r\n" );
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document.write( "Write formulas for areas of triangles BAD' and BAC\r\n" );
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document.write( "    area[BAD'] = 0.5*h*|AD'|,  area [BAC] = 0.5*h*|AC|.\r\n" );
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document.write( "From these formulas, you should see that ratio of the area of triangle BAD' to the area of triangle BAC \r\n" );
document.write( "equals the ratio of the length AD' to the length of AC.\r\n" );
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document.write( "The proof is complete.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Again, the proof is at the introductory Geometry level.\r
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\n" ); document.write( "\n" ); document.write( "I think that it is should be SELF-EVIDENT to any ordinary average level student.\r
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\n" ); document.write( "\n" ); document.write( "If you want I formulate it as a Theorem, here it is:\r
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document.write( "    If two triangles have common altitude, then the ratio of the areas of these triangles\r\n" );
document.write( "    equals to the ratio of the bases of these triangles.\r\n" );
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\n" ); document.write( "\n" ); document.write( "You may attribute this theorem to me and call it \"The Kleyn's theorem\".\r
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\n" ); document.write( "\n" ); document.write( "All Edwin' exclamations in his posts, related to this problem and stating
\n" ); document.write( "that my arguments are incorrect, are erroneous.\r
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\n" ); document.write( "\n" ); document.write( "So, for safety of your mind, ignore all of them.\r
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\n" ); document.write( "\n" ); document.write( "The fact that Edwin still has not cancel these exclamations
\n" ); document.write( "on his own after all our discussions is simply his oversight.\r
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