document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=415334>Question 415334</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This question is from the practice test of the CSET MATH/SCIENCE SUBTEST.\r<BR>\n" );
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document.write( "Four congruent triangles, each having legs of length a and b and hypotenuse of length c, are arranged to produce square EFGH.\r<BR>\n" );
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document.write( "Using your knowledge of algebra and geometry:\r<BR>\n" );
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document.write( "*write an expression for the area of square EFGH in terms of the length of its sides;\r<BR>\n" );
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document.write( "*write an expression for the area of square EFGH in terms of the area of its component parts (i.e., four triangles and a square); and\r<BR>\n" );
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document.write( "*set these two expressions equal and show that this leads to a proof of the Pythagorean theorem. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #291201 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=291201\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "In order for the 4 triangles to form a square, the triangles must be isosceles right triangles.  That is to say <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ =\ b\"> and a hypotenuse of one of the right triangles forms one of the sides of the square.\r<BR>\n" );
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document.write( "The area of the square is then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large c^2\">\r<BR>\n" );
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document.write( "But the area of one of the triangles is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{ab}{2}\">, hence the square, being composed of 4 triangles has an area of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2ab\">\r<BR>\n" );
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document.write( "Since we have established that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ =\ b\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2ab\"> can be written <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2a^2\"> which is to say <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a^2\ +\ a^2\">.  Furthermore, again since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ =\ b\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a^2\ +\ a^2\ =\ a^2\ +\ b^2\">.\r<BR>\n" );
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document.write( "Since both <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a^2\ +\ b^2\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large c^2\"> are expressions for the same area, namely square EFGH, we can equate the two expressions:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ c^2\ =\ a^2\ +\ b^2\">\r<BR>\n" );
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document.write( "Which is the Pythagorean Theorem.\r<BR>\n" );
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document.write( "Unfortunately, this logic does not strictly prove Pythagora's conjecture.  That is because we created a situation at the beginning that required the triangles to be isosceles right triangles.  Hence, this proof is only strictly valid for isosceles right triangles and not right triangles in general.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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