document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1101052>Question 1101052</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Triangle ABC is isosceles with AB=BC=10 units and apex B (1,8). The base AC is parallel to the x-axis. D (1,2) is a point on AC such that BD is perpendicular to AC.\r<BR>\n" );
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document.write( "a. Find the coordinates of A and C.<BR>\n" );
document.write( "b. Find the area of triangle ABC.<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 #715712 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=715712\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here is what the triangle looks like, with all the listed points and line segments:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28540%2C300%2C-8%2C10%2C-1%2C9%2C%0D%0Agrid%281%29%2Cblue%28line%281%2C2%2C1%2C8%29%29%2Cblue%28line%289%2C2%2C-7%2C2%29%29%2C%0D%0Ablue%28line%281%2C8%2C9%2C2%29%29%2Cblue%28line%281%2C8%2C-7%2C2%29%29%2C%0D%0Ablue%28rectangle%281%2C2%2C1.2%2C2.2%29%29%2Clocate%281.1%2C8.5%2CB%29%2C%0D%0Alocate%28-7.4%2C2%2CA%29%2Clocate%289.1%2C2%2CC%29%2Clocate%281.1%2C2%2CD%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "a. The coordinates of A and C, could be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28-7.2%29 ALIGN=MIDDLE  ALT=\"A%28-7.2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%289%2C2%29 ALIGN=MIDDLE  ALT=\"B%289%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "as I labeled them,<BR>\n" );
document.write( "or they could be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28-7.2%29 ALIGN=MIDDLE  ALT=\"B%28-7.2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%289%2C2%29 ALIGN=MIDDLE  ALT=\"A%289%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "I know the coordinates of the points,<BR>\n" );
document.write( "but I do not know if the person who gave those points their names<BR>\n" );
document.write( "started naming them clockwise from bottom left, or counterclockwise from bottom right.<BR>\n" );
document.write( "b. The area of triangle ABC is 48 square units.<BR>\n" );
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document.write( "HOW DO I KNOW:<BR>\n" );
document.write( "I will \"show my work\", although I did all that work in my head.<BR>\n" );
document.write( "I would like to use side AC as the base of the triangle.<BR>\n" );
document.write( "Line segment BD, perpendicular to base AC from vertex B,<BR>\n" );
document.write( "is obviously the altitude with respect to base AC.<BR>\n" );
document.write( "Its length is the height of the triangle.<BR>\n" );
document.write( "Since B and D have both have the same x-coordinate,<BR>\n" );
document.write( "the length of BD is the difference of their y-coordinates.<BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=height=BD=8-2=6 ALIGN=MIDDLE  ALT=\"height=BD=8-2=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Segment BD divides ABC into two congruent right triangles: triangle ADB and triangle CDB.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BD=6 ALIGN=MIDDLE  ALT=\"BD=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > is one leg of right triangle ADB.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BA=10 ALIGN=MIDDLE  ALT=\"BA=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the hypotenuse, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AD=%22%3F%22 ALIGN=MIDDLE  ALT=\"AD=%22%3F%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the other leg.<BR>\n" );
document.write( "Because it is a right triangle we could use the Pythagorean theorem.<BR>\n" );
document.write( "I did not go through that trouble,<BR>\n" );
document.write( "because I recognized triangle ADB as a 6-8-10 right triangle,<BR>\n" );
document.write( "one with sides measuring 6, 9, 10.<BR>\n" );
document.write( "(Teachers like to use 3-4-5, 6-8-10, 9-12-15, ... right triangles<BR>\n" );
document.write( "because they remember that the 3:4:5 ratio gives you a right triangle<BR>\n" );
document.write( "with integer side lengths).<BR>\n" );
document.write( "Using the Pythagorean theorem, I would show my work by writing<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AD%5E2%2BBD%5E2=BA%5E2 ALIGN=MIDDLE  ALT=\"AD%5E2%2BBD%5E2=BA%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AD%5E2%2B6%5E2=10%5E2 ALIGN=MIDDLE  ALT=\"AD%5E2%2B6%5E2=10%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AD%5E2%2B36=100 ALIGN=MIDDLE  ALT=\"AD%5E2%2B36=100\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AD%5E2=100-36 ALIGN=MIDDLE  ALT=\"AD%5E2=100-36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AD%5E2=64 ALIGN=MIDDLE  ALT=\"AD%5E2=64\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AD=sqrt%2864%29 ALIGN=MIDDLE  ALT=\"AD=sqrt%2864%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28AD=4%29 ALIGN=MIDDLE  ALT=\"highlight%28AD=4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
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document.write( "As we know that ADB and CDB are congruent right triangles,<BR>\n" );
document.write( "we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AD=CD ALIGN=MIDDLE  ALT=\"AD=CD\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "because Corresponding Parts of Congruent Triangles are Congruent (which teachers abbreviate as CPCTC).<BR>\n" );
document.write( "So we know that A and C are at 8 unit distance to either side of D,<BR>\n" );
document.write( "on the line parallel to the x-axis that goes through D,<BR>\n" );
document.write( "and that is the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2 ALIGN=MIDDLE  ALT=\"y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, the y-coordinate of A and C is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the x-coordinates of those two points<BR>\n" );
document.write( "have a difference of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the x-coordinate of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%281%2C2%29 ALIGN=MIDDLE  ALT=\"D%281%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . <BR>\n" );
document.write( "Hence, A and C have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=1-8=-7%2C%22and%22%2CX=1%2B8=9%29 ALIGN=MIDDLE  ALT=\"system%28x=1-8=-7%2C%22and%22%2CX=1%2B8=9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "I had to make a choice, so I assigned <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-7 ALIGN=MIDDLE  ALT=\"x=-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > to A and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=9 ALIGN=MIDDLE  ALT=\"x=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > to C.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The area of a triangle is calculated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=base%2Aheight%2F2 ALIGN=MIDDLE  ALT=\"area=base%2Aheight%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so I calculated it as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=%28base%2F2%29%2Aheight=%28AD%29%2A%28BD%29=8%2A6=48 ALIGN=MIDDLE  ALT=\"area=%28base%2F2%29%2Aheight=%28AD%29%2A%28BD%29=8%2A6=48\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "I could have written <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=base=AC=AD%2BBD=8%2B8=16 ALIGN=MIDDLE  ALT=\"base=AC=AD%2BBD=8%2B8=16\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=base%2Aheight%2F2=AC%2ABD%2F2=16%2A6%2F2=48 ALIGN=MIDDLE  ALT=\"area=base%2Aheight%2F2=AC%2ABD%2F2=16%2A6%2F2=48\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but I just calculated it my way in my head so as not to waste time and paper.\r<BR>\n" );
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document.write( "NOTE:<BR>\n" );
document.write( "We did not need all the information given.<BR>\n" );
document.write( "We were told that B is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%281%2C8%29 ALIGN=MIDDLE  ALT=\"B%281%2C8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that:<BR>\n" );
document.write( "1) AC is parallel to the x-axis,<BR>\n" );
document.write( "2) BD is perpendicular to AC, and<BR>\n" );
document.write( "3) the coordinates of D are  and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%281%2C2%29 ALIGN=MIDDLE  ALT=\"D%281%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Any two of those statements allows you to figure out the remaining statement.,<BR>\n" );
document.write( "however, it is nice to have one less thing to figure out. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );