document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=832600>Question 832600</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the slope of the bisector of the angle from L1 to L2. L1 passes through (-2,8) and (6,4), while L2 contains the points (0,12) and (-2,8).. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #502170 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=502170\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> MY WAY:<BR>\n" );
document.write( "A picture is worth a thousand words, but I am very wordy nonetheless.<BR>\n" );
document.write( "We have point A(-2,8), point B(6,4), and point C(-2,8).<BR>\n" );
document.write( "The rays L1 (AB) and L2 (AC) form angle BAC.<BR>\n" );
document.write( "I drew all that in red below, and added a few extra items in green.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-4%2C16%2C-2%2C18%2C%0D%0Agrid%281%29%2Cred%28circle%28-2%2C8%2C0.3%29%29%2C%0D%0Ared%28circle%286%2C4%2C0.3%29%29%2Cred%28circle%280%2C12%2C0.3%29%29%2C%0D%0Agreen%28circle%282%2C16%2C0.3%29%29%2Clocate%28-2.6%2C8%2CA%29%2C%0D%0Alocate%285.4%2C4%2CB%29%2Clocate%280.2%2C12%2CC%29%2C%0D%0Ared%28arrow%28-2%2C8%2C12%2C1%29%29%2Cred%28arrow%28-2%2C8%2C3%2C18%29%29%2C%0D%0Alocate%282.2%2C16%2Cgreen%28D%29%29%2Clocate%283.4%2C10%2Cgreen%28E%29%29%2C%0D%0Agreen%28circle%284%2C10%2C0.3%29%29%2Cgreen%28line%282%2C16%2C6%2C4%29%29%2C%0D%0Agreen%28circle%280%2C8%2C0.3%29%29%2Cgreen%28circle%282%2C8%2C0.3%29%29%2Cgreen%28circle%286%2C8%2C0.3%29%29%2C%0D%0Alocate%280.2%2C8%2CX%29%2Clocate%282.2%2C8%2CY%29%2Clocate%286.2%2C8%2CZ%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Point D(2,16) is on ray L1 because AC and AD have the same slope.<BR>\n" );
document.write( "(It was easy to design it that way by making triangles CAX and DAY similar).<BR>\n" );
document.write( "Point D(2,16) is at the same distance from A as point B.<BR>\n" );
document.write( "(I know that because I made triangles DAY and ABZ congruent).<BR>\n" );
document.write( "BAD is an isosceles triangle.BAD is an isosceles triangle.<BR>\n" );
document.write( "(It is also a right triangle, but I don't care).<BR>\n" );
document.write( "In isosceles triangle BAD, median AE is a median, an altitude, and a bisector of angle BAC.<BR>\n" );
document.write( "(You must have been taught that about isosceles triangles in geometry class).<BR>\n" );
document.write( "The slope of AE can be calculated from the coordinates of A and E.<BR>\n" );
document.write( "E, being the midpoint of BD has the coordinates<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5BE%5D=%28x%5BB%5D%2Bx%5BD%5D%29%2F2=%286%2B2%29%2F2=8%2F2=4 ALIGN=MIDDLE  ALT=\"x%5BE%5D=%28x%5BB%5D%2Bx%5BD%5D%29%2F2=%286%2B2%29%2F2=8%2F2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5BE%5D=%28y%5BB%5D%2By%5BD%5D%29%2F2=%284%2B16%29%2F2=20%2F2=10 ALIGN=MIDDLE  ALT=\"y%5BE%5D=%28y%5BB%5D%2By%5BD%5D%29%2F2=%284%2B16%29%2F2=20%2F2=10\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The slope of AE is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y%5BE%5D-y%5BA%5D%29%2F%28x%5BE%5D-x%5BA%5D%29=%2810-8%29%2F%284-%28-2%29%29=2%2F6=highlight%281%2F3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "ALTERNATE WAYS TO SOLVE IT:\r<BR>\n" );
document.write( "\n" );
document.write( "THE HARDER WAY:<BR>\n" );
document.write( "You could calculate the slope of AB and the slope of AC to get the tangents of the angles AB and AC make with the positive x-axis:<BR>\n" );
document.write( "angle ZAB (measured as a negative angle, swept clockwise from AZ to AB), and<BR>\n" );
document.write( "angle ZAC (measured as a positive angle, swept counterclockwise from AZ to AC).<BR>\n" );
document.write( "You could average those angles to get the angle the bisector makes with the positive x-axis:<BR>\n" );
document.write( "angle ZAE (measured as a positive angle, swept counterclockwise from AZ to AE).<BR>\n" );
document.write( "Slope of AB =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F2=tan%28ZAB%29 ALIGN=MIDDLE  ALT=\"-1%2F2=tan%28ZAB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ZAB=-0.46365 ALIGN=MIDDLE  ALT=\"ZAB=-0.46365\"   DISABLED_style= \"max-width:90%; height:auto;\" >(approximately, in radians) (or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-26.565%5Eo ALIGN=MIDDLE  ALT=\"-26.565%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > approximately, in degrees)<BR>\n" );
document.write( "Slope of AC =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2=tan%28ZAC%29 ALIGN=MIDDLE  ALT=\"2=tan%28ZAC%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ZAC=1.10715 ALIGN=MIDDLE  ALT=\"ZAC=1.10715\"   DISABLED_style= \"max-width:90%; height:auto;\" >(approximately, in radians) (or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=63.435%5Eo ALIGN=MIDDLE  ALT=\"63.435%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > approximately, in degrees)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ZAE=%28ZAB%2BZAC%29%2F2 ALIGN=MIDDLE  ALT=\"ZAE=%28ZAB%2BZAC%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ZAE=0.32175 ALIGN=MIDDLE  ALT=\"ZAE=0.32175\"   DISABLED_style= \"max-width:90%; height:auto;\" >(approximately, in radians) (or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18.435%5Eo ALIGN=MIDDLE  ALT=\"18.435%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > approximately, in degrees)<BR>\n" );
document.write( "Then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28ZAE%29=0.33333 ALIGN=MIDDLE  ALT=\"tan%28ZAE%29=0.33333\"   DISABLED_style= \"max-width:90%; height:auto;\" >= slope of AE (calculated approximsately).<BR>\n" );
document.write( "To calculate an exact value we need to use trigonometric identities to calculate <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28%28ZAB%2BZAC%29%2F2%29 ALIGN=MIDDLE  ALT=\"tan%28%28ZAB%2BZAC%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > directly from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F2=tan%28ZAB%29 ALIGN=MIDDLE  ALT=\"-1%2F2=tan%28ZAB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2=tan%28ZAC%29 ALIGN=MIDDLE  ALT=\"2=tan%28ZAC%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "AN EASIER WAY:<BR>\n" );
document.write( "Realizing that the product of the lopes of AB and AC is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we conclude that BAC is a right angle, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90%5Eo ALIGN=MIDDLE  ALT=\"90%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The angle the bisector makes with the positive x-axis must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=45%5Eo ALIGN=MIDDLE  ALT=\"45%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > less than the angle AC makes with the positive x-axis.<BR>\n" );
document.write( "We know the tangent of the angle AC makes with the positive x-axis,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28ZAC%29=2=slope%28AC%29 ALIGN=MIDDLE  ALT=\"tan%28ZAC%29=2=slope%28AC%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We also know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2845%5Eo%29=1 ALIGN=MIDDLE  ALT=\"tan%2845%5Eo%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the slope of the bisector is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28ZAC-45%5Eo%29=%28tan%28ZAC%29-tan%2845%5Eo%29%29%2F%281%2Btan%28ZAC%29tan%2845%5Eo%29%29=%282-1%29%2F%281%2B2%2A1%29=1%2F%281%2B2%29=highlight%281%2F3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );