document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=959109>Question 959109</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A triangle has coordinates of (1,1), (4,1), and (3,4). is it a 30-60-90 triangle and why? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #586203 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20855)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=586203\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> THERE ARE TWO special triangles in trigonometry. One is the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30 ALIGN=MIDDLE  ALT=\"30\"   DISABLED_style= \"max-width:90%; height:auto;\" >°-<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60 ALIGN=MIDDLE  ALT=\"60\"   DISABLED_style= \"max-width:90%; height:auto;\" >°-<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90 ALIGN=MIDDLE  ALT=\"90\"   DISABLED_style= \"max-width:90%; height:auto;\" >° triangle. The other is the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=isosceles ALIGN=MIDDLE  ALT=\"isosceles\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+right ALIGN=MIDDLE  ALT=\"+right\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+triangle ALIGN=MIDDLE  ALT=\"+triangle\"   DISABLED_style= \"max-width:90%; height:auto;\" >. They are special because, with simple geometry, we can know the ratios of their sides.\r<BR>\n" );
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document.write( "so, what we need to do first is find the sides length using given points and distance formula\r<BR>\n" );
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document.write( "(1,1), (4,1), and (3,4)\r<BR>\n" );
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document.write( "the distance between (1,1)and (4,1) \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%28%284-1%29%5E2%2B%281-1%29%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28%284-1%29%5E2%2B%281-1%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%283%5E2%2B0%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%283%5E2%2B0%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%283%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%283%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=3 ALIGN=MIDDLE  ALT=\"d=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the distance between (1,1)and (3,4) \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%28%283-1%29%5E2%2B%284-1%29%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28%283-1%29%5E2%2B%284-1%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%282%5E2%2B3%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%282%5E2%2B3%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%2813%29 ALIGN=MIDDLE  ALT=\"d=sqrt%2813%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >-exactly\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=3.6 ALIGN=MIDDLE  ALT=\"d=3.6\"   DISABLED_style= \"max-width:90%; height:auto;\" >-approximately\r<BR>\n" );
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document.write( "the distance between (4,1) and (3,4)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%28%284-3%29%5E2%2B%281-4%29%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28%284-3%29%5E2%2B%281-4%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%281%5E2%2B%28-3%29%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%281%5E2%2B%28-3%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%2810%29 ALIGN=MIDDLE  ALT=\"d=sqrt%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >-exactly\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=3.2 ALIGN=MIDDLE  ALT=\"d=3.2\"   DISABLED_style= \"max-width:90%; height:auto;\" >-approximately\r<BR>\n" );
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document.write( "so, the shortest side is  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > units long, then the longer side is  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2810%29 ALIGN=MIDDLE  ALT=\"sqrt%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the longest side is  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2813%29 ALIGN=MIDDLE  ALT=\"sqrt%2813%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "now, check if these are sides of a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30 ALIGN=MIDDLE  ALT=\"30\"   DISABLED_style= \"max-width:90%; height:auto;\" >°-<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60 ALIGN=MIDDLE  ALT=\"60\"   DISABLED_style= \"max-width:90%; height:auto;\" >°-<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90 ALIGN=MIDDLE  ALT=\"90\"   DISABLED_style= \"max-width:90%; height:auto;\" >° triangle\r<BR>\n" );
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document.write( "first, use Pythagorean theorem because triangle has an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90 ALIGN=MIDDLE  ALT=\"90\"   DISABLED_style= \"max-width:90%; height:auto;\" >° angle\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%2813%29%29%5E2=%28sqrt%2810%29%29%5E2%2B3%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%2813%29%29%5E2=%28sqrt%2810%29%29%5E2%2B3%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13=10%2B9 ALIGN=MIDDLE  ALT=\"13=10%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%3C%3E19 ALIGN=MIDDLE  ALT=\"13%3C%3E19\"   DISABLED_style= \"max-width:90%; height:auto;\" >=>your triangle is not a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30 ALIGN=MIDDLE  ALT=\"30\"   DISABLED_style= \"max-width:90%; height:auto;\" >°-<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60 ALIGN=MIDDLE  ALT=\"60\"   DISABLED_style= \"max-width:90%; height:auto;\" >°-<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90 ALIGN=MIDDLE  ALT=\"90\"   DISABLED_style= \"max-width:90%; height:auto;\" >° triangle\r<BR>\n" );
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document.write( "other way is to use a theorem: \r<BR>\n" );
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document.write( " In a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30 ALIGN=MIDDLE  ALT=\"30\"   DISABLED_style= \"max-width:90%; height:auto;\" >°-<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60 ALIGN=MIDDLE  ALT=\"60\"   DISABLED_style= \"max-width:90%; height:auto;\" >°-<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90 ALIGN=MIDDLE  ALT=\"90\"   DISABLED_style= \"max-width:90%; height:auto;\" >° triangle the sides are in the ratio <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%3A+2+%3A+sqrt%28+3%29 ALIGN=MIDDLE  ALT=\"1+%3A+2+%3A+sqrt%28+3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "now, the sides of this triangle are in ratio <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%3Asqrt%2810%29%3Asqrt%2813%29 ALIGN=MIDDLE  ALT=\"3%3Asqrt%2810%29%3Asqrt%2813%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "compared to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%3A+2+%3A+sqrt%28+3%29 ALIGN=MIDDLE  ALT=\"1+%3A+2+%3A+sqrt%28+3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can see that  these ratios are not equivalent; so, your triangle is not a <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30 ALIGN=MIDDLE  ALT=\"30\"   DISABLED_style= \"max-width:90%; height:auto;\" >°-<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60 ALIGN=MIDDLE  ALT=\"60\"   DISABLED_style= \"max-width:90%; height:auto;\" >°-<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90 ALIGN=MIDDLE  ALT=\"90\"   DISABLED_style= \"max-width:90%; height:auto;\" >° triangle \r<BR>\n" );
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document.write( "now, see it on a graph:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-5%2C+5%2C+-5%2C+5%2C%0D%0A%0D%0Acircle%281%2C1%2C.03%29%2Ccircle%284%2C1%2C.03%29%2Ccircle%283%2C4%2C.03%29%2C%0D%0A%0D%0Alocate%281%2C1%2CA%29%2Clocate%284%2C1%2CB%29%2Clocate%283%2C4.3%2CC%29%2C%0D%0A%0D%0Aline%281%2C1%2C4%2C1%29%2Cline%281%2C1%2C3%2C4%29%2Cline%283%2C4%2C4%2C1%29%2C%0D%0A+graph%28+600%2C+600%2C+-5%2C+5%2C+-5%2C+5%2C+0%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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