document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1108272>Question 1108272</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The perimeter of a right triangle is 144 cm. The base is 12 cm. less than the adjacent. Find the area. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #723296 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=723296\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The easiest way to calculate the area of a right triangle is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=%281%2F2%29%2Aleg%5B1%5D%2Aleg%5B2%5D ALIGN=MIDDLE  ALT=\"area=%281%2F2%29%2Aleg%5B1%5D%2Aleg%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "because the legs are perpendicular to each other,<BR>\n" );
document.write( "so one leg can be considered the base, and the other is the height.<BR>\n" );
document.write( "The difficult part is calculating the length of those legs.<BR>\n" );
document.write( "The sides of the triangle measure<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=leg%5B1%5D=36cm ALIGN=MIDDLE  ALT=\"leg%5B1%5D=36cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=leg%5B2%5D=48cm ALIGN=MIDDLE  ALT=\"leg%5B2%5D=48cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=hypotenuse=60cm ALIGN=MIDDLE  ALT=\"hypotenuse=60cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In this case,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=%281%2F2%29%2A%2836cm%29%2A%2848cm%29=highlight%28864cm%5E2%29 ALIGN=MIDDLE  ALT=\"area=%281%2F2%29%2A%2836cm%29%2A%2848cm%29=highlight%28864cm%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "How did I know the leg length?<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE GET RESULTS WITHOUT MUCH WORK WAY:<BR>\n" );
document.write( "Looking at a list of Pythagorean triples,<BR>\n" );
document.write( "or looking at multiples of the popular 3-4-5 triple,<BR>\n" );
document.write( "we figure that 36-48-60 (the 3-4-5 triple times 12)<BR>\n" );
document.write( "add up to 144, and there are differences of 12 between the numbers.<BR>\n" );
document.write( "The sum of the numbers, and differences between numbers in 3-4-5<BR>\n" );
document.write( "are 12 and 1 respectively, so after multiplying everything times 12,<BR>\n" );
document.write( "the sum and differences are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=23%2A12=144 ALIGN=MIDDLE  ALT=\"23%2A12=144\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2A12=12 ALIGN=MIDDLE  ALT=\"1%2A12=12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "THE EXPECTED WAY:<BR>\n" );
document.write( "If there is a figure, it may suggest that<BR>\n" );
document.write( "the \"base\" is one leg of the right triangle.<BR>\n" );
document.write( "If we think the base is the shorter leg,<BR>\n" );
document.write( "\"the adjacent\" can be taken to mean the other leg or the hypotenuse.<BR>\n" );
document.write( "If we think the base is the longer leg, then \"the adjacent\" must be the hypotenuse.<BR>\n" );
document.write( "Either way, what follows is not fun.<BR>\n" );
document.write( "If we think \"the adjacent\" is the hypotenuse,<BR>\n" );
document.write( "the lengths of the sides of the triangle, in cm, are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= hypotenuse,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-12 ALIGN=MIDDLE  ALT=\"x-12\"   DISABLED_style= \"max-width:90%; height:auto;\" >= the \"base\" leg,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E2-%28x-2%29%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%5E2-%28x-2%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=the other leg, and<BR>\n" );
document.write( "the perimeter (in cm) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Bx-12%2Bsqrt%28x%5E2-%28x-12%29%5E2%29=144 ALIGN=MIDDLE  ALT=\"x%2Bx-12%2Bsqrt%28x%5E2-%28x-12%29%5E2%29=144\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-12%2Bsqrt%28x%5E2-%28x%5E2-24x%2B144%29%29=144 ALIGN=MIDDLE  ALT=\"2x-12%2Bsqrt%28x%5E2-%28x%5E2-24x%2B144%29%29=144\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-12%2Bsqrt%28x%5E2-x%5E2%2B24x-144%29=144 ALIGN=MIDDLE  ALT=\"2x-12%2Bsqrt%28x%5E2-x%5E2%2B24x-144%29=144\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-12%2Bsqrt%2824x-144%29=144 ALIGN=MIDDLE  ALT=\"2x-12%2Bsqrt%2824x-144%29=144\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2824x-144%29=144%2B12-2x ALIGN=MIDDLE  ALT=\"sqrt%2824x-144%29=144%2B12-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2824x-144%29=156-2x ALIGN=MIDDLE  ALT=\"sqrt%2824x-144%29=156-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If we get smart, our next step is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sqrt%286x-36%29=156-2x ALIGN=MIDDLE  ALT=\"2sqrt%286x-36%29=156-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "an then we simplify, dividing both sides by 2,to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%286x-36%29=78-x ALIGN=MIDDLE  ALT=\"sqrt%286x-36%29=78-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Squaring both sides of the equal sign, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x-36=6084-156%2Bx%5E2 ALIGN=MIDDLE  ALT=\"6x-36=6084-156%2Bx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-156x-6x%2B6084%2B36=0 ALIGN=MIDDLE  ALT=\"x%5E2-156x-6x%2B6084%2B36=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-162x%2B6120=0 ALIGN=MIDDLE  ALT=\"x%5E2-162x%2B6120=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "AT that point, we may be able to factor it as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-60%29%28x-102%29=0 ALIGN=MIDDLE  ALT=\"%28x-60%29%28x-102%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The solutions to that equation are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=60 ALIGN=MIDDLE  ALT=\"x=60\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=102 ALIGN=MIDDLE  ALT=\"x=102\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We discard <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=102 ALIGN=MIDDLE  ALT=\"x=102\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "because it is not a solution of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%286x-36%29=78-x ALIGN=MIDDLE  ALT=\"sqrt%286x-36%29=78-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "It is just an extraneous solution we introduced when we squared both sides.<BR>\n" );
document.write( "The conclusion is that the hypotenuse measures <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60cm ALIGN=MIDDLE  ALT=\"60cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the base measures <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60cm-12cm=48cm ALIGN=MIDDLE  ALT=\"60cm-12cm=48cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the length of the other leg, in cm, is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E2-%28x-12%29%5E2%29=sqrt%2860%5E2-%2860-12%29%5E2%29=sqrt%283600%2A48%5E2%29=sqrt%281296%29=36 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "If there is another way that is simple, let me know. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );