document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196827>Question 1196827</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A right triangle has a hypotenuse of 1. One leg is x. The other leg is (x - 7).<BR>\n" );
document.write( "Find the area of the triangle.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #829844 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</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=829844\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Edit: Disregard my entire solution, since the premise is based on a typo in the original question posted. Refer to the solution @ikleyn has posted.\r<BR>\n" );
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document.write( "Use the pythagorean theorem <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2+=+c%5E2 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2+=+c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to form the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B%28x-7%29%5E2+=+1%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B%28x-7%29%5E2+=+1%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "That equation solves to these complex values<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%287-i%2Asqrt%2847%29%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%287-i%2Asqrt%2847%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%287%2Bi%2Asqrt%2847%29%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%287%2Bi%2Asqrt%2847%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Use the quadratic formula. <BR>\n" );
document.write( "This shows that we don't get a real number for x and therefore such a right triangle isn't possible.\r<BR>\n" );
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document.write( "Or you could use the triangle inequality theorem<BR>\n" );
document.write( "a = x<BR>\n" );
document.write( "b = x-7<BR>\n" );
document.write( "c = 1\r<BR>\n" );
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document.write( "This theorem says the sum of any two sides must exceed the third side if we want a triangle to be possible<ul><li>a+b > c leads to x+x-7 > 1 which solves to x > 4</li><li>a+c > b leads to x+1 > x-7 which leads to 1 > -7; while it's nice we get a true inequality statement, it's not particularly useful here, so we'll move on</li><li>b+c > a leads to x-7+1 > x which leads to -6 > 0 which is false</li></ul>The <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Bc+%3E+a ALIGN=MIDDLE  ALT=\"b%2Bc+%3E+a\"   DISABLED_style= \"max-width:90%; height:auto;\" > case leads to a contradiction.<BR>\n" );
document.write( "We have the two sides b = x-7 and c = 1 combine to x-6 which is NOT larger than the third side a = x\r<BR>\n" );
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document.write( "Let's replace x with an actual number greater than 7<BR>\n" );
document.write( "Let's go for x = 10<BR>\n" );
document.write( "So,<BR>\n" );
document.write( "a = x = 10<BR>\n" );
document.write( "b = x-7 = 10-7 = 3<BR>\n" );
document.write( "c = 1\r<BR>\n" );
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document.write( "Then notice<BR>\n" );
document.write( "b+c = 3+1 = 4 is NOT larger than a = 10<BR>\n" );
document.write( "Try this out using slips of paper of lengths 10, 3 and 1 (inches, cm, or whatever your favorite unit is)<BR>\n" );
document.write( "You'll find a triangle is impossible to form here. \r<BR>\n" );
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document.write( "Here's an interactive GeoGebra tool to play with<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.geogebra.org/m/yzv48tkk\">https://www.geogebra.org/m/yzv48tkk</a><BR>\n" );
document.write( "If the applet doesn't load, then hover your mouse over the center and click on the curved arrow. It should refresh. If not then let me know and I'll try to fix the issue the best I can.\r<BR>\n" );
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document.write( "In that applet, we have the three sides {1,3,10}<BR>\n" );
document.write( "segment AB = 10<BR>\n" );
document.write( "segment AD = 1<BR>\n" );
document.write( "segment BC = 3<BR>\n" );
document.write( "The dashed circles show all possible locations the points D and B could go. In other words, segment AD swings around point A; while segment BC swings around point B.\r<BR>\n" );
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document.write( "No matter how hard we try, there is no way to join up points C and D to form a triangle. The sides AD = 1 and BC = 3 are simply too short to add to something larger than AB = 10<BR>\n" );
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