document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1031262>Question 1031262</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equations of the bisectors of the interior angles of the triangle whose vertices are (0,4), (-4,-4) and (6,1).\r<BR>\n" );
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document.write( "Find the equation of the line that bisects the acute angle formed by the following lines.<BR>\n" );
document.write( "a.) x-y=0 and x=0<BR>\n" );
document.write( "b.) 7x-y=5 and y=x+1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #646121 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=646121\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the points be P(0,4), Q(-4,-4) and R(6,1).\r<BR>\n" );
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document.write( "Then line PQ will have slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BPQ%5D+=+2 ALIGN=MIDDLE  ALT=\"m%5BPQ%5D+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the equation would be 2x - y + 4 = 0.<BR>\n" );
document.write( "The line QR will have slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BQR%5D+=+1%2F2 ALIGN=MIDDLE  ALT=\"m%5BQR%5D+=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the equation would be x - 2y - 4 = 0.<BR>\n" );
document.write( "Then line PR will have slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BPR%5D+=+-1%2F2 ALIGN=MIDDLE  ALT=\"m%5BPR%5D+=+-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the equation would be x + 2y -8 = 0.\r<BR>\n" );
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document.write( "For the angle bisector at angle Q, let (x,y) be any point on it.  Then the distance of (x,y) from line PQ is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-y%2B4%29%2Fsqrt%285%29 ALIGN=MIDDLE  ALT=\"%282x-y%2B4%29%2Fsqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, while its distance from the line QR is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28x-2y-4%29%2Fsqrt%285%29 ALIGN=MIDDLE  ALT=\"-%28x-2y-4%29%2Fsqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  (The negative sign denotes the fact that points (x,y) above the line x - 2y - 4 = 0 will give a net sign of negative for the expression x-2y-4 upon substitution.)\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-y%2B4%29%2Fsqrt%285%29+=+-%28x-2y-4%29%2Fsqrt%285%29 ALIGN=MIDDLE  ALT=\"%282x-y%2B4%29%2Fsqrt%285%29+=+-%28x-2y-4%29%2Fsqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "==> 2x-y+4 = -x+2y+4  ==> 3x = 3y, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=y%29 ALIGN=MIDDLE  ALT=\"highlight%28x=y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "For the angle bisector at angle P, let (x,y) be any point on it.  Then the distance of (x,y) from line PQ is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-y%2B4%29%2Fsqrt%285%29 ALIGN=MIDDLE  ALT=\"%282x-y%2B4%29%2Fsqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, while its distance from the line PR is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28x%2B2y-8%29%2Fsqrt%285%29 ALIGN=MIDDLE  ALT=\"-%28x%2B2y-8%29%2Fsqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  (The negative sign denotes the fact that points (x,y) below the line x + 2y - 8 = 0 will give a net sign of negative for the expression x+2y-8 upon substitution.)\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-y%2B4%29%2Fsqrt%285%29+=+-%28x%2B2y-8%29%2Fsqrt%285%29 ALIGN=MIDDLE  ALT=\"%282x-y%2B4%29%2Fsqrt%285%29+=+-%28x%2B2y-8%29%2Fsqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "==> 2x -y +4 = -x-2y+8 ==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%283x+%2B+y+-+4+=+0%29 ALIGN=MIDDLE  ALT=\"highlight%283x+%2B+y+-+4+=+0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "By a similar procedure, it easily found that the angle bisector for angle R is simply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y+=+1%29 ALIGN=MIDDLE  ALT=\"highlight%28y+=+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "a.) x-y=0 and x=0.<BR>\n" );
document.write( "Let (x,y) be in the angle bisector with vertex at (0,0).  <BR>\n" );
document.write( "The distance of (x,y) from the line x=0 (the y-axis) is x, while the distance of (x,y) from the line x - y=0 is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28x-y%29%2Fsqrt%282%29 ALIGN=MIDDLE  ALT=\"-%28x-y%29%2Fsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  The negative sign is for the fact that upon substitution of the coordinates of (x,y) into x - y, the sign of the expression is negative.\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-%28x-y%29%2Fsqrt%282%29 ALIGN=MIDDLE  ALT=\"x+=+-%28x-y%29%2Fsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28%28sqrt%282%29%2B1%29x+=+y%29 ALIGN=MIDDLE  ALT=\"highlight%28%28sqrt%282%29%2B1%29x+=+y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, after simplification.\r<BR>\n" );
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document.write( "b.) I leave up to you.\r<BR>\n" );
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