document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=967262>Question 967262</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can an isosceles right triangle have all sides with integer length? Why or why not? \r<BR>\n" );
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document.write( "I Googled this question and received the answer no with an explanation I did not understand. I want to know if you agree that the answer is no and if so why, and if not. I'm having a hard time with geometry concepts so please \"dummy down the answer\".\r<BR>\n" );
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document.write( "I greatly appreciate the help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #591187 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=591187\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> not sure what you are asking.\r<BR>\n" );
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document.write( "an isosceles triangle can have sides that are integers.\r<BR>\n" );
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document.write( "no problem there.\r<BR>\n" );
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document.write( "can it have all 3 sides of integer length.\r<BR>\n" );
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document.write( "i don't see why not.\r<BR>\n" );
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document.write( "the only requirements is that two of the sides have the same length.\r<BR>\n" );
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document.write( "we'll now construct a triangle with two sides of equal length of 7.\r<BR>\n" );
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document.write( "we can make the third side any length we want as long as the third side is not greater than or equal to 14.\r<BR>\n" );
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document.write( "if it was, then that would violate one of the properties of a triangle that the sum of the lengths of any two sides of the triangle must be greater than the length of the third side.\r<BR>\n" );
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document.write( "so our only restriction, if two of the sides are 7, is that the third side has to be less than 14.\r<BR>\n" );
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document.write( "the other restriction is that the length of the third side is greater than the difference between the lengths of the other two sides.\r<BR>\n" );
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document.write( "since the difference between the two sides of 7 each is 0, this means that the third side has to have a length greater than 0.\r<BR>\n" );
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document.write( "that means the third side can be any number between 0 and 14, but not including 0 or 14.\r<BR>\n" );
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document.write( "there are 13 integers between 0 and 14 and not including 0 and 14.\r<BR>\n" );
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document.write( "those integers are 1,2,3,4,5,6,7,8,9,10,11,12,13.\r<BR>\n" );
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document.write( "the third side can be any one of those.\r<BR>\n" );
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document.write( "the base angles of the isosceles triangle will be different, depending on the length of the third side.\r<BR>\n" );
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document.write( "since it is an isosceles triangle, then two sides are equal and the two base angles are equal.\r<BR>\n" );
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document.write( "the third angle of the triangle is the vertex of the triangle which is the angle between the two equal sides.\r<BR>\n" );
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document.write( "BUT YOU SAID AN ISOSCELES \"RIGHT\" TRIANGLE.\r<BR>\n" );
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document.write( "THAT'S DIFFERENT.\r<BR>\n" );
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document.write( "CONTINUE READING TO SEE WHY THE THIRD SIDE CANNOT BE AN INTEGER WHEN THE ISO9SCELES TRIANGLE IS A RIGHT TRIANGLE.\r<BR>\n" );
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document.write( "if it's an isosceles right triangle then the hypotenuse of the right triangle is the base of the triangle.\r<BR>\n" );
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document.write( "now there is a restriction.\r<BR>\n" );
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document.write( "the vertex angle has to be 90 degrees for the right triangle to be an isosceles right triangle.\r<BR>\n" );
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document.write( "the side of the triangle that is not one of the equal sides of the triangle is the hypotenuse of this right triangle.\r<BR>\n" );
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document.write( "the formula to find the length of the hypotenuse of a right triangle is:\r<BR>\n" );
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document.write( "c^2 = a^2 + b^2\r<BR>\n" );
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document.write( "c is the hypotenuse<BR>\n" );
document.write( "a and b are the legs of the right triangole.\r<BR>\n" );
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document.write( "since the triangle is an isosceles right triangle, then the legs are equal to each other.\r<BR>\n" );
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document.write( "this means that x = b, so you can replace b in the formula with a because b is the same length as a.\r<BR>\n" );
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document.write( "the formuls of c^2 = a^2 + b^2 becomes c^2 = a^2 + a^2 which becomes c^2 = 2 * a^2.\r<BR>\n" );
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document.write( "you want to solve for c which is the length of the hypotenuse.\r<BR>\n" );
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document.write( "take the square root of both sides of the equation of c^2 = 2 * a^2 and you will get:\r<BR>\n" );
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document.write( "sqrt(c^2) = sqrt(2 * a^2)\r<BR>\n" );
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document.write( "sqrt(c^2) is equal to c.\r<BR>\n" );
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document.write( "formula becomes c = sqrt(2 * a^2)\r<BR>\n" );
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document.write( "since sqrt(2 * a^2) is equivalent to sqrt(2) * sqrt(a^2), then the equation becomes:\r<BR>\n" );
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document.write( "c = sqrt(2) * sqrt(a^2).\r<BR>\n" );
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document.write( "since sqrt(a^2) is equal to a, the formula becomes:\r<BR>\n" );
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document.write( "c = sqrt(2) * a\r<BR>\n" );
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document.write( "the square root of 2 is an irrational number, so sqrt(2) * a can never be a rational number.\r<BR>\n" );
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document.write( "since an integer is a special form of a ration number where the denominator is equal to 1, this means that sqrt(2) * a can never be an integer.\r<BR>\n" );
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document.write( "the answer provided to you is correct.\r<BR>\n" );
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document.write( "the third side of an isosceles right triangle can never be an integer.\r<BR>\n" );
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document.write( "let's take our triangle and swee how it works in numbers.\r<BR>\n" );
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document.write( "the equal legs are 7 apiece.\r<BR>\n" );
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document.write( "it's a right triangle so the hypotenuse squared is equal to 2 * 7^2\r<BR>\n" );
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document.write( "the hypotenuse is equal to sqrt (2 * 7^2) which is equal to sqrt(2) * sqrt(7^2) which is equal to sqrt(2) * 7.\r<BR>\n" );
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document.write( "sqrt(2) is not a rational number, so sqrt(2) * 7 can never be a rational number.\r<BR>\n" );
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document.write( "no cigars.\r<BR>\n" );
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document.write( "third side can't be a rational number.\r<BR>\n" );
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document.write( "therefore the third side can't be an integer because an integer is a special form of a rational number.\r<BR>\n" );
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document.write( "a rational number is defined as a fraction where the numerator and the denominator are both integers.\r<BR>\n" );
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document.write( "3/4 is a rational number because 3 and 4 are integers.\r<BR>\n" );
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document.write( "5 is a rational number because 5 is equivalent to 5/1 and 5 and 1 are integers.\r<BR>\n" );
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document.write( "sqrt(2) is not a rational number and so the third side can't be an integer because the third side of the isosceles right triangle will always be equal to any number * sqrt(2).\r<BR>\n" );
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document.write( " </SPAN>\n" );
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