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 #851518 by mccravyedwin(408)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Ikleyn has made a false assumption, when she said:\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( "...is nothing else [but] the ratio of the length AD' to AC. \r\n" );
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document.write( "While it is true that a PERPENDICULAR drawn from a vertex to the opposite\r\n" );
document.write( "side of a triangle divides the opposite side into the same ratio as the ratio\r\n" );
document.write( "of the areas of the right triangles formed by the perpendicular, this is NOT\r\n" );
document.write( "true when the line is not perpendicular to the opposite side.\r\n" );
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document.write( "Ikleyn was apparently thinking of what would have been the case if BD' had been\r\n" );
document.write( "perpendicular to AC, which it is not.  \r\n" );
document.write( "She was thinking of this case:\r\n" );
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document.write( "We see from this, that the ratio she assumed would be correct.\r\n" );
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document.write( "\"y%2F%28y%2Bz%29+=+%28expr%281%2F2%29%2Ax%2Ay%29%2F%28expr%281%2F2%29%2A%28y%2Bz%29%2Ax%29\"\r\n" );
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document.write( "But that only holds when the line from the vertex to the opposite side is\r\n" );
document.write( "perpendicular to the opposite side.  But in the original figure BD' is not\r\n" );
document.write( "perpendicular to AC.\r\n" );
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document.write( "The only way to do the problem is to find the actual areas and compare them.  \r\n" );
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document.write( "Ikleyn has used the law of cosines to find the cosine to be that of a special\r\n" );
document.write( "angle, 120o.  So we could then use the sine of that special angle and the area \r\n" );
document.write( "formula using the sine to find the area of triangle ABC.\r\n" );
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document.write( "Then we could use the law of cosines again on triangle ABD' to find side AD'.  \r\n" );
document.write( "Then we could use the area formula using the sine again to find the area of\r\n" );
document.write( "triangle ABD'.\r\n" );
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document.write( "If Ikleyn wants to correct her post, I will then delete this one, and make\r\n" );
document.write( "another post using a different screen name to inform the student that the correct\r\n" );
document.write( "solution is now online, and which tutor gave it.\r\n" );
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document.write( "I'll give Ikleyn a couple of days to do this.  If she hasn't done it by Monday,\r\n" );
document.write( "I will delete this, and post the correct solution under a different screen name.\r\n" );
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document.write( "Edwin

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