document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203533>Question 1203533</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A triangle has the coordinates (2, 7), (3, 5) and (1,0)\r<BR>\n" );
document.write( "\n" );
document.write( "Is this a right triangle? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #839172 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3835)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=839172\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "<font color=red size=4>Answer:</font> No, this is not a right triangle.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Explanation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Label the points A,B,C. The order of which doesn't matter.<BR>\n" );
document.write( "A = (2,7)<BR>\n" );
document.write( "B = (3,5)<BR>\n" );
document.write( "C = (1,0)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I'll use the distance formula to compute how far it is from A to B. <BR>\n" );
document.write( "This will give us the length of segment AB.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A = (x1,y1) = (2,7) and B = (x2,y2) = (3,5)<BR>\n" );
document.write( "d = distance from A to B = length of segment AB<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+%282-3%29%5E2+%2B+%287-5%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%282-3%29%5E2+%2B+%287-5%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+%28-1%29%5E2+%2B+%282%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%28-1%29%5E2+%2B+%282%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+1+%2B+4+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+1+%2B+4+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+5+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+5+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+2.236068 ALIGN=MIDDLE  ALT=\"d+=+2.236068\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Segment AB is exactly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29 ALIGN=MIDDLE  ALT=\"sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > units long, which approximates to about 2.236068 units.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Use that same formula to find these other segment lengths:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC+=+sqrt%2829%29+=+5.385165 ALIGN=MIDDLE  ALT=\"BC+=+sqrt%2829%29+=+5.385165\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC+=+sqrt%2850%29+=+5%2Asqrt%282%29+=+7.071068 ALIGN=MIDDLE  ALT=\"AC+=+sqrt%2850%29+=+5%2Asqrt%282%29+=+7.071068\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The decimal values are approximate.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Once we know the side lengths, we can plug them into the pythagorean theorem equation. <BR>\n" );
document.write( "If we get the same thing on both sides, then we conclude the triangle is a right triangle.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%281%2C2%2C%22first%22%2C%22leg%22%29%29%5E2%2B%28matrix%281%2C2%2C%22second%22%2C%22leg%22%29%29%5E2=%28hypotenuse%29%5E2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28AB%29%5E2%2B%28BC%29%5E2=%28AC%29%5E2 ALIGN=MIDDLE  ALT=\"%28AB%29%5E2%2B%28BC%29%5E2=%28AC%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > The hypotenuse is <u><font color=red>always</font></u> the longest side. In this case, the longest side is AC = sqrt(50) = 7.071068 roughly.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%285%29%29%5E2%2B%28sqrt%2829%29%29%5E2=%28sqrt%2850%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%285%29%29%5E2%2B%28sqrt%2829%29%29%5E2=%28sqrt%2850%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2B29=50 ALIGN=MIDDLE  ALT=\"5%2B29=50\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=34=50 ALIGN=MIDDLE  ALT=\"34=50\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Because we ended up with a false equation, it means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28AB%29%5E2%2B%28BC%29%5E2=%28AC%29%5E2 ALIGN=MIDDLE  ALT=\"%28AB%29%5E2%2B%28BC%29%5E2=%28AC%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is also false. <BR>\n" );
document.write( "Furthermore, we do NOT have a right triangle.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Here's another approach that might be more simple depending on your viewpoint.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Once again we have these three given points<BR>\n" );
document.write( "A = (2,7)<BR>\n" );
document.write( "B = (3,5)<BR>\n" );
document.write( "C = (1,0)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's compute the slope of line AB.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A = (x1,y1) = (2,7) and B = (x2,y2) = (3,5)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+slope+=+rise%2Frun+=+matrix%281%2C3%2C%22change%22%2C%22in%22%2C%22y%22%29%2Fmatrix%281%2C3%2C%22change%22%2C%22in%22%2C%22x%22%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28y%5B2%5D+-+y%5B1%5D%29%2F%28x%5B2%5D+-+x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"m+=+%28y%5B2%5D+-+y%5B1%5D%29%2F%28x%5B2%5D+-+x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%285+-+7%29%2F%283+-+2%29 ALIGN=MIDDLE  ALT=\"m+=+%285+-+7%29%2F%283+-+2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28-2%29%2F%281%29 ALIGN=MIDDLE  ALT=\"m+=+%28-2%29%2F%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+-2 ALIGN=MIDDLE  ALT=\"m+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The slope of line AB is -2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If you use the slope formula for the other sides, then:<BR>\n" );
document.write( "slope of BC = 5/2 = 2.5<BR>\n" );
document.write( "slope of AC = 7\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then multiply each possible slope pairing.<BR>\n" );
document.write( "(slope AB)*(slope BC) = (-2)*(2.5) = -5<BR>\n" );
document.write( "(slope AB)*(slope AC) = (-2)*(7) = -14<BR>\n" );
document.write( "(slope BC)*(slope AC) = (2.5)*(7) = 17.5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "None of the pairs multiply to -1. <BR>\n" );
document.write( "Therefore, none of the pairs of lines are perpendicular and we do NOT have a right triangle.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There might be other approaches. Feel free to explore other alternatives.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );