document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=633638>Question 633638</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A point labelled E is drawn on the interior of a square and lines drawn to it from each of the corners. Give an argument that would convince someone else who has studied mathematics that the combined area of the 2 darker shaded triangular regions is half the area of the surrounding square...it may not be { in which case you would have to provide an argument that justifies this conclusion.\r<BR>\n" );
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document.write( "(darker shades are covered by the stars*)<BR>\n" );
document.write( "blank space is covered with dots\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #399042 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=399042\">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( "Construct a square of side measure <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\"> in the first quadrant of the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \mathbb{R}^2\"> plane such that one vertex of the square is at the origin and the two sides adjacent to that vertex are cooincident with the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\"> axes.\r<BR>\n" );
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document.write( "By definition of a square, the four vertices of the square will be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE A(0,0)\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE B(a, 0)\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE C(a,a)\">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE D(0,a)\">\r<BR>\n" );
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document.write( "Let point E be represented by the ordered pair <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (x,y)\">\r<BR>\n" );
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document.write( "The altitude of triangle ABE to side AB is simply the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\"> coordinate of point E, namely <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\">.  The measure of the base of triangle ABE is the distance from <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (0,0)\"> to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (a,0)\">, which is simply <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\">.  The area of a triangle is given by one-half the altitude times the base, or in the case of triangle ABE, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{ay}{2}\">\r<BR>\n" );
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document.write( "Likewise it can be shown that the area of triangle CDE is given by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{a(a\ -\ y)}{2}\ =\ \frac{a^2\ -\ ay}{2}\">\r<BR>\n" );
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document.write( "The sum of these two areas is then:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{ay}{2}\ +\ \frac{a^2\ -\ ay}{2}\ =\ \frac{ay\ +\ a^2\ -\ ay}{2}\ =\ \frac{a^2}{2}\">\r<BR>\n" );
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document.write( "But the area of the square is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a^2\">, so the sum of the areas of the two shaded triangles is in fact one-half of the area of the square.\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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