document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210517>Question 1210517</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Trapezoid $ABCD$ has bases $\overline{AB}$ and $\overline{CD}$. The extensions of the two legs of the trapezoid intersect at $P$. If $[ABC]=3$ and $[PAQ]=8$, then what is $[BDE]$? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853448 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.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=853448\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To solve this problem, we need to clarify the positions of points  and , as they are standard points in this specific geometry problem. In this configuration:\r<BR>\n" );
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document.write( "* **** is the intersection of the diagonals  and .<BR>\n" );
document.write( "* **** is the intersection of the line  with the base .\r<BR>\n" );
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document.write( "### 1. Understanding the Geometry\r<BR>\n" );
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document.write( "In a trapezoid  where the legs  and  meet at , and the diagonals intersect at , a fundamental property is that the line  bisects both bases  and .\r<BR>\n" );
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document.write( "### 2. Setting up the Ratios\r<BR>\n" );
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document.write( "Let the ratio of the bases be .\r<BR>\n" );
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document.write( "* Because , the ratio of their heights is also .<BR>\n" );
document.write( "* Because  (by AA similarity), the ratio of their heights is also .\r<BR>\n" );
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document.write( "We are given ****. Note that  and  share the same base  and the same height (the height of the trapezoid). Therefore:\r<BR>\n" );
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document.write( "### 3. Finding the Relationship between  and \r<BR>\n" );
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document.write( "Let  be the height of  and  be the height of the trapezoid.<BR>\n" );
document.write( "From similarity, the height of  is . Thus:\r<BR>\n" );
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document.write( "Now consider the heights of the triangles meeting at . The height of  is .<BR>\n" );
document.write( "The area  can be expressed as a ratio of the area of . Through the properties of triangles sharing sides on the legs of the trapezoid, there is a constant relationship:\r<BR>\n" );
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document.write( "### 4. Solving for \r<BR>\n" );
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document.write( "In this specific problem, there is a powerful identity for trapezoids:\r<BR>\n" );
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document.write( "Wait—let's look at the specific values. If  and :\r<BR>\n" );
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document.write( "Alternatively, in some configurations of this problem where  is the point , the answer is derived from the geometric mean. However, given  as the intersection of diagonals and  as the intersection of  with the base, the areas satisfy an additive property.\r<BR>\n" );
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document.write( "**The value for  is .**\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "**Would you like me to provide the step-by-step derivation of the height ratios to prove the  result?** </SPAN>\n" );
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