document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=965826>Question 965826</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the area of quadrilateral ABCD whose sides are 9m 40m 28m 15m and angle ABC=90° </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #590320 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><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=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=590320\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find the area of quadrilateral ABCD whose sides are 9m 40m 28m 15m and angle ABC=90°<BR>\n" );
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document.write( "Assuming AB = 9 and BC = 40:<BR>\n" );
document.write( "Put A at the Origin, B at (0,9), C at (40,9)<BR>\n" );
document.write( "Point D is the intersection of 2 circles, one centered at A with r = 15 and one centered at C with r = 28.<BR>\n" );
document.write( "--> x^2 + y^2 = 225 and<BR>\n" );
document.write( "(x-40)^2 + (y-9)^2 = 784<BR>\n" );
document.write( "Find the intersection of the 2 circles:<BR>\n" );
document.write( "x^2 + y^2 = 225<BR>\n" );
document.write( "x^2 + y^2 - 80x - 18y = -897<BR>\n" );
document.write( "------------------------------------- subtract<BR>\n" );
document.write( "80x + 18y = 1122 is the equation of the line passing thru the 2 intersections of the circles.<BR>\n" );
document.write( "--> y = (561 - 40x)/9<BR>\n" );
document.write( "x^2 + y^2 = 225<BR>\n" );
document.write( "Sub for y<BR>\n" );
document.write( "x^2 + ((561 - 40x)/9)^2 = 225<BR>\n" );
document.write( "81x^2 + (561 - 40x)^2 = 225*81 = 18225<BR>\n" );
document.write( "81x^2 + 1600x^2 - 44880x + 314721 = 18225<BR>\n" );
document.write( "1681x^2 - 44880x + 296496 = 0<BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=solve_quadratic_equation.solver>SOLVE quadratic equation (work shown, graph etc)</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1681x%5E2%2B-44880x%2B296496+=+0 ALIGN=MIDDLE  ALT=\"1681x%5E2%2B-44880x%2B296496+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-44880%29%5E2-4%2A1681%2A296496=20575296 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-44880%29%5E2-4%2A1681%2A296496=20575296\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=20575296 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28--44880%2B-sqrt%28+20575296+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--44880%2B-sqrt%28+20575296+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%28-44880%29%2Bsqrt%28+20575296+%29%29%2F2%5C1681+=+14.6983938132064 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-44880%29%2Bsqrt%28+20575296+%29%29%2F2%5C1681+=+14.6983938132064\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28-44880%29-sqrt%28+20575296+%29%29%2F2%5C1681+=+12 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-44880%29-sqrt%28+20575296+%29%29%2F2%5C1681+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1681x%5E2%2B-44880x%2B296496 ALIGN=MIDDLE  ALT=\"1681x%5E2%2B-44880x%2B296496\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1681x%5E2%2B-44880x%2B296496+=+%28x-14.6983938132064%29%2A%28x-12%29 ALIGN=MIDDLE  ALT=\"1681x%5E2%2B-44880x%2B296496+=+%28x-14.6983938132064%29%2A%28x-12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 14.6983938132064, 12.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1681%2Ax%5E2%2B-44880%2Ax%2B296496+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1681%2Ax%5E2%2B-44880%2Ax%2B296496+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
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document.write( "==============<BR>\n" );
document.write( "x = 14.6984<BR>\n" );
document.write( "--> y = -2.9929<BR>\n" );
document.write( "D is (14.6984,-2.9929)<BR>\n" );
document.write( "======================<BR>\n" );
document.write( "Write the points in order.  I'll use (14.7,-3) for D<BR>\n" );
document.write( "A...B...C...D....A<BR>\n" );
document.write( "0...0..40..14.7..0<BR>\n" );
document.write( "0...9...9..-3....0<BR>\n" );
document.write( "Add the diagonal products starting at the upper left.<BR>\n" );
document.write( "0+0-120+0 = 120<BR>\n" );
document.write( "Add the diagonal products starting at the lower left.<BR>\n" );
document.write( "0+360+132.3+0 = 492.3<BR>\n" );
document.write( "------------<BR>\n" );
document.write( "Find the difference = 492.3 - 120 = 372.3<BR>\n" );
document.write( "The area is 1/2 that<BR>\n" );
document.write( "Area = 186.15 sq units<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );