document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=122779>Question 122779</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum of the lengths of one side of a right triangle and the hypotenuse is 20cm. The other leg is 5cm shorter than the hypotenuse.  Find the length of each side.\r<BR>\n" );
document.write( "\n" );
document.write( "x^2+(x-5)^2=20^2<BR>\n" );
document.write( "x^2+x^2-25x+25=400<BR>\n" );
document.write( "2x^2-25x-375=0<BR>\n" );
document.write( "x^2-5x-178.50=0\r<BR>\n" );
document.write( "\n" );
document.write( "I just want to double check my steps to make sure I did this right.  Thanks is advance. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #90157 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=90157\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The way I solved this problem was to start with three equations.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Start by letting x and y equal the two legs of the triangle and h equal the hypotenuse.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "The first sentence in the problem says the sum of one side and the hypotenuse equals 20 cm.<BR>\n" );
document.write( "In equation form this can be written as:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2B+h+=+20 ALIGN=MIDDLE  ALT=\"x+%2B+h+=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Solve this equation for x and you have:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+20+-+h ALIGN=MIDDLE  ALT=\"x+=+20+-+h\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "The next sentence in the problem says the other leg of the triangle is 5 cm less than the<BR>\n" );
document.write( "hypotenuse. In equation form this is:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+h+-+5 ALIGN=MIDDLE  ALT=\"y+=+h+-+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "The final equation in the series comes from the Pythagorean theorem which for this problem<BR>\n" );
document.write( "says that the sum of the squares of the legs equals the square of the hypotenuse. In<BR>\n" );
document.write( "this problem it is:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+y%5E2+=+h%5E2 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+y%5E2+=+h%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Substituting 20 - h for x and h - 5 for y results in:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2820+-+h%29%5E2+%2B+%28h+-+5%29%5E2+=+h%5E2 ALIGN=MIDDLE  ALT=\"%2820+-+h%29%5E2+%2B+%28h+-+5%29%5E2+=+h%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Squaring out the two terms on the left side gives:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=400+-+40h+%2B+h%5E2+%2B+h%5E2+-+10h+%2B+25+=+h%5E2 ALIGN=MIDDLE  ALT=\"400+-+40h+%2B+h%5E2+%2B+h%5E2+-+10h+%2B+25+=+h%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Combining the like terms on the left side results in:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2h%5E2+-+50h+%2B+425+=+h%5E2 ALIGN=MIDDLE  ALT=\"2h%5E2+-+50h+%2B+425+=+h%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Subtract h^2 from both sides and you get the standard quadratic form of:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+-+50h+%2B+425+=+0 ALIGN=MIDDLE  ALT=\"h%5E2+-+50h+%2B+425+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "If you apply the quadratic formula to get the solutions to this problem, you start with:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"h+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "and substitute 1 for a, -50 for b, and 425 for c to get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+%28-%28-50%29+%2B-+sqrt%28+%28-50%29%5E2-4%2A1%2A425+%29%29%2F%282%2A1%29+ ALIGN=MIDDLE  ALT=\"h+=+%28-%28-50%29+%2B-+sqrt%28+%28-50%29%5E2-4%2A1%2A425+%29%29%2F%282%2A1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "In the radical <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-50%29%5E2+-4%2A1%2A425+=+2500+-+1700+=+800 ALIGN=MIDDLE  ALT=\"%28-50%29%5E2+-4%2A1%2A425+=+2500+-+1700+=+800\"   DISABLED_style= \"max-width:90%; height:auto;\" > and substituting this gives:<BR>\n" );
document.write( ".\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+%28-%28-50%29+%2B-+sqrt%28+800%29%29%2F%282%2A1%29+ ALIGN=MIDDLE  ALT=\"h+=+%28-%28-50%29+%2B-+sqrt%28+800%29%29%2F%282%2A1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "The -(-50) = +50. The square root of 800 = 28.28427125. And the denominator is just 2. These<BR>\n" );
document.write( "simplifications result in:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+%2850+%2B-+28.28427125%29%2F2 ALIGN=MIDDLE  ALT=\"h+=+%2850+%2B-+28.28427125%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Dividing the two terms in the numerator by the denominator results in:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+25+%2B-+14.14213562 ALIGN=MIDDLE  ALT=\"h+=+25+%2B-+14.14213562\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "This means that there are two possible answers for h. The first is 25 + 14.14213562 <BR>\n" );
document.write( "which is 39.14213562. However, this cannot be the answer because we know that x + h must<BR>\n" );
document.write( "be 20. But if h is 39.14213562 then x has to be a negative number if the sum of x and h<BR>\n" );
document.write( "is to be 20.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "So the next trial is to assume the answer is h = 25 - 14.14213562 and this subtraction leads<BR>\n" );
document.write( "to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "h = 10.85786438 cm<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Then we can use the equation x + h = 20 to find x. Substitute 10.85786438 for h and x becomes<BR>\n" );
document.write( "defined by the equation:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2B+10.85786438+=+20 ALIGN=MIDDLE  ALT=\"x+%2B+10.85786438+=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Subtract 10.85786438 from both sides of this equation and the result is:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+20+-+10.85786438+=+9.14213562 ALIGN=MIDDLE  ALT=\"x+=+20+-+10.85786438+=+9.14213562\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "So one leg is 9.14213562 cm.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "The other leg of the triangle (side y) is 5 cm smaller than h and this is:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+h+-+5+=+10.85786438+-+5+=+5.85786438 ALIGN=MIDDLE  ALT=\"y+=+h+-+5+=+10.85786438+-+5+=+5.85786438\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "So the length of the final side is 5.85786438 cm.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "In summary, the three sides are: hypotenuse = 10.85786438 cm, leg x = 9.14213562 cm, and<BR>\n" );
document.write( "leg y = 5.85786438 cm.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "If you check these values out you will find that they satisfy all three equations.<BR>\n" );
document.write( "In other words:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "x + h = 9.14213562 + 10.85786438 = 20<BR>\n" );
document.write( ".<BR>\n" );
document.write( "and:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "y = h - 5 <BR>\n" );
document.write( ".<BR>\n" );
document.write( "5.85786432 = 10.85786438 - 5<BR>\n" );
document.write( ".<BR>\n" );
document.write( "and <BR>\n" );
document.write( ".<BR>\n" );
document.write( "x^2 + y^2 = h^2 <BR>\n" );
document.write( ".<BR>\n" );
document.write( "(9.14213562)^2 + (5.85786432)^2 = (10.85786438)^2<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Hope this helps you to see how to set up and solve this problem.<BR>\n" );
document.write( ".<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );