document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=675205>Question 675205</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The hypotenuse of a right triangle is 9cm less than two times the shortest leg.<BR>\n" );
document.write( "The longer leg is 36cm. Find the length of the shortest leg of the right triangle and find the hypotenuse.<BR>\n" );
document.write( "Please show all steps.<BR>\n" );
document.write( "Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #419757 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=419757\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let x = the length of the shortest leg. Then the length of the hypotenuse would be: 2x - 9.<br><BR>\n" );
document.write( "We have expressions for all three sides of a right triangle. So they should fit in the Pythagorean equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29%5E2+%2B+%2836%29%5E2+=+%282x-9%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%29%5E2+%2B+%2836%29%5E2+=+%282x-9%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now we solve this for x. First we simplify:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+1296+=+4x%5E2-36x%2B81 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+1296+=+4x%5E2-36x%2B81\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we want one side to be zero. Subtracting both terms of the left side we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+3x%5E2-36x-1215 ALIGN=MIDDLE  ALT=\"0+=+3x%5E2-36x-1215\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we factor (or use the Quadratic Formula). We can factor out the GCF of 3:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+3%28x%5E2-12x-405%29 ALIGN=MIDDLE  ALT=\"0+=+3%28x%5E2-12x-405%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The second factor will factor, but not easily. (So I understand if you prefer to use the Quadratic Formula instead.)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+3%28x-27%29%28x%2B15%29 ALIGN=MIDDLE  ALT=\"0+=+3%28x-27%29%28x%2B15%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From the Zero Product Property we know that one of these factors must be zero. The \"3\" will not be zero but the other two could:<BR>\n" );
document.write( "x - 27 = 0 or x + 15 = 0<BR>\n" );
document.write( "Solving there we get:<BR>\n" );
document.write( "x = 27 or x = -15<br><BR>\n" );
document.write( "Since x represents the shortest leg of our triangle, we will reject the negative answer. So the shortest leg is 27.<br><BR>\n" );
document.write( "And we can use this value for x to find the hypotenuse:<BR>\n" );
document.write( "2(27) - 9<BR>\n" );
document.write( "54 - 9<BR>\n" );
document.write( "45<br><BR>\n" );
document.write( "So the shortest leg is 27 cm and the hypotenuse is 54 cm.<br><BR>\n" );
document.write( "P.S. This problem can be solved very quickly if you are familiar with 3/4/5 right triangles. Not only will triangles with sides of 3, 4 and 5 be right triangles, so will any triangle that has multiples of 3, 4 and 5 for its sides.<br><BR>\n" );
document.write( "Since the longer leg given to you, 36, is a multiple of 4 (the longer leg of 3/4/5 triangles), you might see if the other legs are the same multiples. 36 is 9 * 4. So let's pretend that the short leg is 9 * 3 = 27 and the hypotenuse is 9 * 5 = 45. All we have to do now is make sure that the hypotenuse is \"9cm less than two times the shortest leg\":<BR>\n" );
document.write( "Is 2*27 - 9 = 45? Answer: Yes! So we could solve this very quickly this way. (NOTE: If the hypotenuse had not been \"9cm less than two times the shortest leg\" then we would have to solve this using algebra like the first, longer solution above.) </SPAN>\n" );
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