document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203372>Question 1203372</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In trapezoid ABCD segments AB and CD are parallel. Point P is the intersection of diagonals AC and BD. The area of APAB is 16 square units, and the area of ▲PCD is 25 square units. What is the area of trapezoid ABCD?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #838880 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=838880\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "There's a strange typo when you wrote \"The area of APAB is 16 square units\".<BR>\n" );
document.write( "I have a feeling it should read \"The area of triangle PAB is 16 square units\".<BR>\n" );
document.write( "The copy/paste system probably confused the triangle symbol for the letter A. But it doesn't really explain why the triangle symbol shows up next to \"PCD\".\r<BR>\n" );
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document.write( "For many visual learners, such as myself, it helps to draw out a diagram. I often use GeoGebra.<BR>\n" );
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document.write( "Claim: Triangles PAB and PCD are similar. \r<BR>\n" );
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document.write( "How can we prove this claim? Well we just need two pairs of congruent angles.\r<BR>\n" );
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document.write( "One pair is angle APB = angle CPD, which are vertical angles.<BR>\n" );
document.write( "Vertical angles are always congruent.\r<BR>\n" );
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document.write( "Then another pair is that<BR>\n" );
document.write( "angle ABP = angle CDP<BR>\n" );
document.write( "these are alternate interior angles. Such angles are congruent when we have parallel lines.\r<BR>\n" );
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document.write( "We will then use the angle angle (AA) similarity theorem to conclude that triangle PAB is similar to triangle PCD. <BR>\n" );
document.write( "The order of the lettering is important so we know how the corresponding angles pair up.\r<BR>\n" );
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document.write( "We are given <BR>\n" );
document.write( "area(PAB) = 16<BR>\n" );
document.write( "area(PCD) = 25<BR>\n" );
document.write( "The ratio of those areas is 16:25<BR>\n" );
document.write( "The square root of each part gives 4:5<BR>\n" );
document.write( "This operation is valid because we have shown triangle PAB is similar to triangle PCD.<BR>\n" );
document.write( "If two linear pieces are in ratio m:n, then their areas are in ratio m^2:n^2. This applies to similar figures only.\r<BR>\n" );
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document.write( "So if PB = 4 for instance, then its corresponding paired side PD would be 5 units long.\r<BR>\n" );
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document.write( "The thing is that we don't know how long PB is<BR>\n" );
document.write( "Let's call it x.<BR>\n" );
document.write( "PD must then be (5/4)x units long so PB and PD are in ratio 4:5\r<BR>\n" );
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document.write( "This bit of scratch work might help show why that is.<BR>\n" );
document.write( "PB : PD<BR>\n" );
document.write( "x : (5/4)x<BR>\n" );
document.write( "4x : 5x<BR>\n" );
document.write( "4 : 5<BR>\n" );
document.write( "I multiplied both sides by 4 to get rid of the fraction.\r<BR>\n" );
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document.write( "Rotate the diagram so that segment BP is completely horizontal. <BR>\n" );
document.write( "Triangles PAB and PAD have base PB and PD respectively. <BR>\n" );
document.write( "Both triangles PAB and PAD have the same height h.\r<BR>\n" );
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document.write( "area(PAB) = 0.5*base*height<BR>\n" );
document.write( "area(PAB) = 0.5*x*h<BR>\n" );
document.write( "and<BR>\n" );
document.write( "area(PAD) = 0.5*base*height<BR>\n" );
document.write( "area(PAD) = 0.5*(5/4)x*h<BR>\n" );
document.write( "area(PAD) = (5/4)*(0.5*x*h)<BR>\n" );
document.write( "area(PAD) = (5/4)*area(PAB)\r<BR>\n" );
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document.write( "Whatever triangle PAB's area is, we multiply by 5/4 to get the area of triangle PAD.\r<BR>\n" );
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document.write( "But we were given PAB to be 16 square units.<BR>\n" );
document.write( "area(PAD) = (5/4)*area(PAB)<BR>\n" );
document.write( "area(PAD) = (5/4)*16<BR>\n" );
document.write( "area(PAD) = 20\r<BR>\n" );
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document.write( "Follow a similar set of steps to conclude that<BR>\n" );
document.write( "area(PCB) = (5/4)*area(PAB)<BR>\n" );
document.write( "area(PCB) = (5/4)*16<BR>\n" );
document.write( "area(PCB) = 20\r<BR>\n" );
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document.write( "We have all the pieces we need to find the area of the trapezoid<BR>\n" );
document.write( "area(ABCD) = area(PAB) + area(PCD) + area(PCB) + area(PAD)<BR>\n" );
document.write( "area(ABCD) = 16 + 25 + 20 + 20<BR>\n" );
document.write( "area(ABCD) = 81\r<BR>\n" );
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document.write( "Answer:  <font color=red size=4>81 square units</font><BR>\n" );
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