document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=179578>Question 179578</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Homework Question:\r<BR>\n" );
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document.write( "Find the area of the triangle enclosed by the lines,  <BR>\n" );
document.write( "1 .     y=2/3x -7/3<BR>\n" );
document.write( "2.        y =-1/2x+3/2<BR>\n" );
document.write( "3.       y=-6x-5<BR>\n" );
document.write( "Find the points of intersection and then map on a graph.\r<BR>\n" );
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document.write( "I started out by finding the x and y coordinates of each line to plot on the graph.  The triangle I get is so small that I know I need to find the points of intersection but I am not sure how.  Then I      guess find the side length I think and lastly the area but unsure of what path to take.   I am a middle grade teacher working towards my endorsement and taking class.  This was the homework the teacher gave.  He made it up, it did not come from a text.<BR>\n" );
document.write( "I would appreciate any help you can give.<BR>\n" );
document.write( "Thanks!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #134710 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=134710\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here are the lines graphed. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+300%2C+-5%2C+5%2C+-5%2C+5%2C+%282%2F3%29x-%287%2F3%29%2C+-%281%2F2%29x%2B3%2F2%2C%0D%0A-6x-5%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To find the points of intersection, set the equations equal to each other.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=%282%2F3%29x-%287%2F3%29 ALIGN=MIDDLE  ALT=\"y%5B1%5D=%282%2F3%29x-%287%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D=-%281%2F2%29x%2B3%2F2 ALIGN=MIDDLE  ALT=\"y%5B2%5D=-%281%2F2%29x%2B3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B3%5D=-6x-5 ALIGN=MIDDLE  ALT=\"y%5B3%5D=-6x-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That is,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=y%5B2%5D ALIGN=MIDDLE  ALT=\"y%5B1%5D=y%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=y%5B3%5D ALIGN=MIDDLE  ALT=\"y%5B1%5D=y%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D=y%5B3%5D ALIGN=MIDDLE  ALT=\"y%5B2%5D=y%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Here's <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=y%5B2%5D ALIGN=MIDDLE  ALT=\"y%5B1%5D=y%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F3%29x-%287%2F3%29=-%281%2F2%29x%2B3%2F2 ALIGN=MIDDLE  ALT=\"%282%2F3%29x-%287%2F3%29=-%281%2F2%29x%2B3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Solve for x, then go to eq. 1 or eq. 2 and solve for y.<BR>\n" );
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document.write( "Next you need to decide how you'll calculate the area : Heron's formula or the traditional base-height relationship.<BR>\n" );
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document.write( "Heron's formula:<BR>\n" );
document.write( "You can find all the lengths of the triangle using the distance formula,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=sqrt%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2 ALIGN=MIDDLE  ALT=\"D=sqrt%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and then use Heron's formula for triangle area.<BR>\n" );
document.write( "Heron's formula uses the three sides of a triangle a,b,c and the semi-perimeter,s.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=sqrt%28s%2A%28s-a%29%2A%28s-b%29%2A%28s-c%29%29 ALIGN=MIDDLE  ALT=\"A=sqrt%28s%2A%28s-a%29%2A%28s-b%29%2A%28s-c%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s=%28a%2Bb%2Bc%29%2F2 ALIGN=MIDDLE  ALT=\"s=%28a%2Bb%2Bc%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The traditional area of a triangle would give you,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%281%2F2%29bh ALIGN=MIDDLE  ALT=\"A=%281%2F2%29bh\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where b stands for base and h is the height, the perpendicular distance from the base to opposite base vertex (altitude).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+300%2C+300%2C+-5%2C+5%2C+-5%2C+5%2Cgrid%28+1+%29%2C%0D%0Acircle%28+3.28%2C+-0.14%2C+.2+%29%2C+%0D%0Acircle%28+-1.18%2C+2.09%2C+.2+%29%2C+%0D%0Acircle%28+-0.4%2C+-2.6%2C+.2+%29%2C+%0D%0Agreen%28line%28-0.4%2C-2.6%2C3.28%2C-0.14%29%29%2C%0D%0Aline%283.28%2C-0.14%2C-1.18%2C2.09%29%2C%0D%0Aline%28-1.18%2C2.09%2C-0.4%2C-2.6%29%29%0D%0A%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Once you have the three intersection points then you can call one side the base, let's assume the green line is the base. <BR>\n" );
document.write( "Find its length with the distance formula. <BR>\n" );
document.write( "Then you would need to find the altitude shown here as the red line.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+300%2C+300%2C+-5%2C+5%2C+-5%2C+5%2Cgrid%28+1+%29%2C%0D%0Acircle%28+3.28%2C+-0.14%2C+.2+%29%2C+%0D%0Acircle%28+-1.18%2C+2.09%2C+.2+%29%2C+%0D%0Acircle%28+-0.4%2C+-2.6%2C+.2+%29%2C+%0D%0Ared%28line%28-1.18%2C2.09%2C1.2%2C-1.5%29%29%2C%0D%0Acircle%281.2%2C-1.5%2C0.2%29%2C%0D%0Agreen%28line%28-0.4%2C-2.6%2C3.28%2C-0.14%29%29%2C%0D%0Aline%283.28%2C-0.14%2C-1.18%2C2.09%29%2C%0D%0Aline%28-1.18%2C2.09%2C-0.4%2C-2.6%29%29%0D%0A%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since you know the slope of the green line, you also know the slope of the red line since perpendicular lines follow this relationship,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5B1%5Dm%5B2%5D=-1 ALIGN=MIDDLE  ALT=\"m%5B1%5Dm%5B2%5D=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You can then formulat the equation of the line using this slope and the intersection point opposite of the base (which you calculated previously) using the point-slope form of a line,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B4%5D-y%5Bp%5D=m%5B2%5D%28x-x%5Bp%5D%29 ALIGN=MIDDLE  ALT=\"y%5B4%5D-y%5Bp%5D=m%5B2%5D%28x-x%5Bp%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We'll call that equation 4.<BR>\n" );
document.write( "Now you need to find the intersection point of line 4 and the green base line. <BR>\n" );
document.write( "Again set the two equations equal to each other and solve for x and y. <BR>\n" );
document.write( "Now that you have the intersection point, use the distance formula to find the length of the altitude.<BR>\n" );
document.write( "Then use the area formula,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%281%2F2%29bh ALIGN=MIDDLE  ALT=\"A=%281%2F2%29bh\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
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