document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=985127>Question 985127</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How to find the Maximum Area of the Rectangle inscribed in a Right Triangle of sides a, b and Hypotenuse length c. Please I want a Algebra Proof, not one using Calculus or Derivatives. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #605985 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=605985\">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( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=solver91311&illustration_name=rectintri\">.\r<BR>\n" );
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document.write( "Refer to the figure.\r<BR>\n" );
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document.write( "The area of right triangle ABC is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{1}{2}ab\">. But the area of the right triangle is also the sum of the areas of the two triangles formed by the diagonal of the inscribed rectangle, namely triangle CEB and CEA.  The side of the rectangle that measures <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> is the altitude of triangle CEB and the side of the rectangle that measures <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> is the altitude of triangle CEA.  So the areas of these two triangles are given by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{1}{2}ax\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{1}{2}by\">.  Hence, the following relationship holds for any arbitrary right triangle with an inscribed rectangle:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{1}{2}ab\ =\ \frac{1}{2}ax\ +\ \frac{1}{2}by\">\r<BR>\n" );
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document.write( "Solving for y:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ a\ -\ \frac{ax}{b}\">\r<BR>\n" );
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document.write( "(you can verify the algebra for yourself)\r<BR>\n" );
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document.write( "Then, since the area of a rectangle is given by the length times the width, the area of the rectangle as a function of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A(x)\ =\ ax\ -\ \frac{ax^2}{b}\">\r<BR>\n" );
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document.write( "or, in standard form:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A(x)\ =\ -\ \frac{ax^2}{b}\ +\ ax\">\r<BR>\n" );
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document.write( "Note that the graph of this function is a parabola that opens downward because of the negative lead coefficient.\r<BR>\n" );
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document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinate of the vertex of a parabola of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \rho(x)\ =\ \alpha{x^2}\ +\ \beta{x}\ +\ \gamma\"> is given by\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x_v\ =\ \frac{-\beta}{2\alpha}\">\r<BR>\n" );
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document.write( "For your area function, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \beta\ =\ a\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \alpha\ =\ -\frac{a}{b}\">, hence the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinate of the vertex, and therefore the point where the maximum value of the area function will be found is at:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x_{max}\ =\ \frac{-a}{\frac{-2a}{b}}\ =\ \frac{b}{2}\">\r<BR>\n" );
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document.write( "The maximum rectangle is formed when the sides of the rectangle are exactly one-half of the legs of the right triangle.\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\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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