document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210173>Question 1210173</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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<BR>\n" );
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document.write( "The sides of triangle ABC are 3, 5, and 7. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851447 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851447\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step by step.\r<BR>\n" );
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document.write( "**1. Geometric Interpretation**\r<BR>\n" );
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document.write( "We are looking for the probability that BD < 4. This is equivalent to finding the area of the region inside triangle ABC where BD < 4, and dividing it by the total area of triangle ABC.\r<BR>\n" );
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document.write( "**2. Triangle ABC and Ray BP**\r<BR>\n" );
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document.write( "* Let P be a random point inside triangle ABC.<BR>\n" );
document.write( "* Ray BP intersects AC at point D.<BR>\n" );
document.write( "* We want to find the probability that BD < 4.\r<BR>\n" );
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document.write( "**3. Law of Cosines**\r<BR>\n" );
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document.write( "Let's find the angles of triangle ABC using the Law of Cosines:\r<BR>\n" );
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document.write( "* Let a = 3, b = 5, c = 7.<BR>\n" );
document.write( "* $\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{25 + 49 - 9}{2(5)(7)} = \frac{65}{70} = \frac{13}{14}$<BR>\n" );
document.write( "* $\cos B = \frac{a^2 + c^2 - b^2}{2ac} = \frac{9 + 49 - 25}{2(3)(7)} = \frac{33}{42} = \frac{11}{14}$<BR>\n" );
document.write( "* $\cos C = \frac{a^2 + b^2 - c^2}{2ab} = \frac{9 + 25 - 49}{2(3)(5)} = \frac{-15}{30} = -\frac{1}{2}$\r<BR>\n" );
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document.write( "Since $\cos C = -1/2$, angle C = 120 degrees.\r<BR>\n" );
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document.write( "**4. Area of Triangle ABC**\r<BR>\n" );
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document.write( "We can use Heron's formula to find the area of triangle ABC.\r<BR>\n" );
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document.write( "* Semi-perimeter: $s = (3 + 5 + 7)/2 = 15/2 = 7.5$<BR>\n" );
document.write( "* Area = $\sqrt{s(s-a)(s-b)(s-c)} = \sqrt{7.5(4.5)(2.5)(0.5)} = \sqrt{42.1875} = \frac{15\sqrt{3}}{4}$\r<BR>\n" );
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document.write( "**5. Geometric Reasoning**\r<BR>\n" );
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document.write( "* Let's consider the locus of points P inside triangle ABC such that BD = 4.<BR>\n" );
document.write( "* We can create a triangle ABD' where BD' = 4.<BR>\n" );
document.write( "* The locus of P such that BD = 4 is a line parallel to AD'.<BR>\n" );
document.write( "* The region where BD < 4 is the region inside triangle ABC that is closer to B than to D'.\r<BR>\n" );
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document.write( "**6. Ratio of Areas**\r<BR>\n" );
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document.write( "* The probability we are looking for is the ratio of the area of the region where BD < 4 to the area of the triangle ABC.<BR>\n" );
document.write( "* Let's consider a point P that makes BD = 4.<BR>\n" );
document.write( "* Let's determine the ratio of AD/CD.<BR>\n" );
document.write( "* By the Angle Bisector Theorem, if P were to lie on an angle bisector, we could find the ratios. However, P is not necessarily on an angle bisector.<BR>\n" );
document.write( "* The ratio AD/CD is dependent on the position of point P.\r<BR>\n" );
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document.write( "**7. Difficulties**\r<BR>\n" );
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document.write( "* Determining the exact shape of the region where BD < 4 is challenging without further geometric constructions and calculations.<BR>\n" );
document.write( "* The problem requires a more advanced approach, potentially involving barycentric coordinates or other geometric methods.\r<BR>\n" );
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document.write( "**8. Conclusion**\r<BR>\n" );
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document.write( "Without further information or a more advanced approach, it's difficult to find the exact probability. However, we know that the probability is the ratio of the area of the region where BD < 4 to the total area of the triangle.\r<BR>\n" );
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document.write( "Unfortunately, I don't see a way to solve this problem without more advanced techniques.<BR>\n" );
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