document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1158596>Question 1158596</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The line x+2y = 5 divides the circle x2 +y2 = 25 into two arcs. Calculate their lengths. The interior of the circle is divided into two regions by the line. Calculate their areas. Give three significant digits for your answers. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #781557 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=781557\">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( "The first thing that you must accomplish is to find the coordinates of the two points of intersection between the given circle and the given secant line.\r<BR>\n" );
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document.write( "There are several approaches, but I chose this one:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ +\ 2y\ =\ 5\ \Right\ x\ =\ -2y\ +\ 5\">\r<BR>\n" );
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document.write( "And\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^2\ +\ y^2\ =\ 25\ \Right\ x^2\ =\ -y^2\ +\ 25\">\r<BR>\n" );
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document.write( "So\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (-2y\ +\ 5)^2\ =\ -y^2\ +\ 25\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 5y^2\ -\ 20y\ =\ 0\">\r<BR>\n" );
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document.write( "And I leave it as an exercise for the student to verify that this leads to the conclusion that the points of intersection are <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \(-3,4\)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \(5,0\)\">\r<BR>\n" );
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document.write( "The formula for the measure of an arc of a circle is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s\ =\ r\theta\"> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \theta\"> is the angle subtended measured in radians.  Our next challenge is to find the value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \theta\"> for the arc <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large ACB\">.  See figure:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=solver91311&illustration_name=Line_CircleIntersectionCroppeda.jpg\">\r<BR>\n" );
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document.write( "By inspection, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sin\theta\ =\ \frac{4}{5}\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \cos\theta\ =\ -\frac{3}{5}\">, therefore <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \tan\theta\ =\ -\frac{3}{4}\">\r<BR>\n" );
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document.write( "But since the range of the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large arctan\"> function is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large -\frac{\pi}{2}\ <\ \theta\ <\ \frac{\pi}{2}\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \arctan\({-\frac{3}{4}\)\"> yields a negative angle in QIV.  So the measure of the desired angle is given by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \theta\ =\ \arctan\({-\frac{3}{4}\)\ +\ \pi\">\r<BR>\n" );
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document.write( "And then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ s_{ACB}\ =\ 5\theta\">\r<BR>\n" );
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document.write( "Since the major arc, namely <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large BDA\">, is simply the rest of the circle,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ s_{BDA}\ =\ 10\pi\ -\ s_{ACB}\">\r<BR>\n" );
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document.write( "The arithmetic, as is my habit, is left to you.\r<BR>\n" );
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document.write( "Given the above solution to the arc length part of the problem, you should be able to solve the area part by use of the formula for the area of a circle, which I expect that you already know, and the formula for the area of a sector which is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A\ =\ \frac{1}{2}r^2\theta\">\r<BR>\n" );
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document.write( "Don't forget to set your calculator to radian mode, don't round anything off until you get a final answer to a calculation, and never carry a rounded answer into a subsequent calculation.<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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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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