document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1099314>Question 1099314</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine graphically,the co-ordinates of the vertices of th triangle bounded by 2x-3y+6=0;2x+3y-18=0;y-2=0 Also find the area of this triangle<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 #713758 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=713758\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Determining graphically means we have to start by graphing.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y-2=0%29 ALIGN=MIDDLE  ALT=\"blue%28y-2=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y=2%29 ALIGN=MIDDLE  ALT=\"blue%28y=2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is easy to graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-4%2C6%2C-2%2C8%2Cgrid%281%29%2C%0D%0Ablue%28line%28-6%2C2%2C7%2C2%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The other two lines can be esily graphed using their x- and y-intercepts.<BR>\n" );
document.write( "Once the intercepts for a line are found,<BR>\n" );
document.write( "we just plot the points, and draw a line through them.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%282x-3y%2B6=0%29 ALIGN=MIDDLE  ALT=\"red%282x-3y%2B6=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%282x-3y=-6%29 ALIGN=MIDDLE  ALT=\"red%282x-3y=-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "has x- and y-intercepts (-3,0) and (0,2) respectively,<BR>\n" );
document.write( "that can be found by susbtituting 0 for y and x respectively.<BR>\n" );
document.write( "The x- and y-intercepts for<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%282x%2B3y-18=0%29 ALIGN=MIDDLE  ALT=\"green%282x%2B3y-18=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%282x%2B3y=18%29 ALIGN=MIDDLE  ALT=\"green%282x%2B3y=18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "can be similarly found to be (9,0) and (0,6) .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C300%2C-5%2C10%2C-2%2C8%2Cgrid%281%29%2C%0D%0Ablue%28line%28-6%2C2%2C11%2C2%29%29%2C%0D%0Ared%28line%28-6%2C-2%2C9%2C8%29%29%2C%0D%0Agreen%28line%2812%2C-2%2C-3%2C8%29%29%2C%0D%0Ared%28circle%28-3%2C0%2C0.2%29%29%2Cred%28circle%280%2C2%2C0.2%29%29%2C%0D%0Agreen%28circle%289%2C0%2C0.2%29%29%2Cgreen%28circle%280%2C6%2C0.2%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "From the graph it looks like the vertices of the triangle are (0,2) , (3,4) , and (6,2) .<BR>\n" );
document.write( "With the graph, it is easy to see that triangle as one with<BR>\n" );
document.write( "a horizontal base of length <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=6 ALIGN=MIDDLE  ALT=\"b=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and a vertical height <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=2 ALIGN=MIDDLE  ALT=\"h=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Then, the area of the triangle can be calculated as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2A2%2F2=6%2A2%2F2=Highlight%286%29 ALIGN=MIDDLE  ALT=\"b%2A2%2F2=6%2A2%2F2=Highlight%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "DO WE NEED TO REMEMBER THE FORMULA FOR THE AREA OF A TRIANGLE, AND DO THE CALCULATION?<BR>\n" );
document.write( "Not really. It is obvious that the triangle is half of the rectangle outlined below, so we could just count squares: one rectangle = 12 squares, one hald rectangle = 6 squares.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C300%2C-5%2C10%2C-2%2C8%2Cgrid%281%29%2C%0D%0Ablue%28line%28-6%2C2%2C11%2C2%29%29%2C%0D%0Ared%28line%28-6%2C-2%2C9%2C8%29%29%2C%0D%0Agreen%28line%2812%2C-2%2C-3%2C8%29%29%2C%0D%0Ablue%28line%280%2C2%2C6%2C2%29%29%2Cblue%28line%280%2C2%2C0%2C4%29%29%2C%0D%0Ablue%28line%280%2C4%2C6%2C4%29%29%2Cblue%28line%286%2C2%2C6%2C4%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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document.write( "WHAT IF THE TEACHER WANTS YOU TO SHOW MORE WORK?<BR>\n" );
document.write( "I see two places where you would have to show that you really understand how to figure out the problem:<BR>\n" );
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document.write( "1) Any side of a triangle (a segment between two vertices) can be called the base, and then the perpendicular distance to the other vertex is the height.<BR>\n" );
document.write( "In this case it is easiest to call the horizontal segment between (0,2)  and (6,2) the base of the triangle,<BR>\n" );
document.write( "and then the vertical distance from that base to point (3,4) would be the height.<BR>\n" );
document.write( "Theoretically you would calculate the length of the horizontal base as the difference in x-coordinates of the endpoints of that base, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=base=6-0=6 ALIGN=MIDDLE  ALT=\"base=6-0=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Theoretically you would calculate the vertical height as the difference in y-coordinates between the base, and the other vertex, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=height+=4-2=2 ALIGN=MIDDLE  ALT=\"height+=4-2=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "2) Would your teacher insist that you verify the coordinates with some calculations?<BR>\n" );
document.write( "Maybe we should really verify that the lines intersect at points with exactly those coordinates,<BR>\n" );
document.write( "by substituting those coordinates into the equations of the lines,<BR>\n" );
document.write( "because sometimes things are not quite exactly the way they look.<BR>\n" );
document.write( "We do not need to verify (0,2) ,<BR>\n" );
document.write( "which we have shown above that is a point that<BR>\n" );
document.write( "is part of blue line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y=2%29 ALIGN=MIDDLE  ALT=\"blue%28y=2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "is part of red line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%282x-3y=-6%29 ALIGN=MIDDLE  ALT=\"red%282x-3y=-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=3%2Cy=4%29 ALIGN=MIDDLE  ALT=\"system%28x=3%2Cy=4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the coordinates of (3,4),<BR>\n" );
document.write( "we see that it is indeed a solution of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28red%282x-3y=-6%29%2Cgreen%282x%2B3y-18=0%29%29 ALIGN=MIDDLE  ALT=\"system%28red%282x-3y=-6%29%2Cgreen%282x%2B3y-18=0%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=6%2Cy=2%29 ALIGN=MIDDLE  ALT=\"system%28x=6%2Cy=2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the coordinates of (6,2),<BR>\n" );
document.write( "we see that it is indeed a solution of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28blue%28y=2%29%2Cgreen%282x%2B3y-18=0%29%29 ALIGN=MIDDLE  ALT=\"system%28blue%28y=2%29%2Cgreen%282x%2B3y-18=0%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
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