document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206736>Question 1206736</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what is the area of the largest semi-circle which can be inscribed within rhombus abcd with edge length 2 cm if the measure of angle a is 60 degrees </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #844365 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13198)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=844365\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "I will take a crack at this....<br><BR>\n" );
document.write( "Without proof, it seems to me that the largest area will be if the diameter of the semicircle is parallel to the long diagonal of the rhombus.  We get a picture like this....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-1%2C3%2C-1%2C3%0D%0A%2Cline%280%2C0%2C1%2Csqrt%283%29%29%2Cline%281%2Csqrt%283%29%2C3%2Csqrt%283%29%29%2Cline%283%2Csqrt%283%29%2C2%2C0%29%2Cline%282%2C0%2C0%2C0%29%0D%0A%2Cred%28line%282%2F3%2C0%2C8%2F3%2C2sqrt%283%29%2F3%29%29%0D%0A%2Cred%28line%285%2F3%2Csqrt%283%29%2F3%2C2%2F3%2C2sqrt%283%29%2F3%29%29%2Cred%28line%285%2F3%2Csqrt%283%29%2F3%2C5%2F3%2Csqrt%283%29%29%29%0D%0A%2Clocate%280%2C0%2CA%29%2Clocate%280.9%2C1.9%2CB%29%2Clocate%283%2C1.9%2CC%29%2Clocate%282%2C0%2CD%29%0D%0A%2Clocate%280.5%2C1.2%2CE%29%2Clocate%285%2F3%2C1.9%2CF%29%2Clocate%288%2F3%2C2sqrt%283%29%2F3%2CG%29%2Clocate%282%2F3%2C0%2CH%29%0D%0A%2Clocate%285%2F3%2Csqrt%283%29%2F3%2CO%29%0D%0A%2Cgreen%28line%282%2F3%2C0%2C2%2F3%2C2sqrt%283%29%2F3%29%29%0D%0A%2Cgreen%28line%282%2F3%2C2sqrt%283%29%2F3%2C5%2F3%2Csqrt%283%29%29%29%0D%0A%2Cgreen%28line%285%2F3%2Csqrt%283%29%2C8%2F3%2C2sqrt%283%29%2F3%29%29%0D%0A%2Cblue%28line%281%2Csqrt%283%29%2C7%2F6%2C5sqrt%283%29%2F6%29%29%0D%0A%2Clocate%287%2F6%2C5sqrt%283%29%2F6%2CK%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In the figure....<br><BR>\n" );
document.write( "ABCD is the rhombus with side length 2<br><BR>\n" );
document.write( "E, F, G, and H are the points on the rhombus where the inscribed semicircle touches.  GH is the diameter of the semicircle; O is the center of the semicircle; OE, OF, OG, and OH are radii  (I don't know how to draw the semicircle....)<br><BR>\n" );
document.write( "OE is perpendicular to AB and OF is perpendicular to BC (radii to the points of tangency are perpendicular to the sides of the rhombus)<br><BR>\n" );
document.write( "BK is the perpendicular bisector of EF.<br><BR>\n" );
document.write( "With angle A being 60 degrees, triangles AEH, EKB, FBK, and FCG are 30-60-90 right triangles, and triangles HEO, EFO, and OFG are equilateral triangles. <br><BR>\n" );
document.write( "With that picture, the side length of each of the equilateral triangles, and thus the radius of the inscribed semicircle, is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sqrt%283%29%2F3 ALIGN=MIDDLE  ALT=\"2sqrt%283%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The area of the semicircle is then<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%281%2F2%29%28pi%29%28r%5E2%29=%281%2F2%29%28pi%29%2812%2F9%29=%282%2F3%29pi ALIGN=MIDDLE  ALT=\"A=%281%2F2%29%28pi%29%28r%5E2%29=%281%2F2%29%28pi%29%2812%2F9%29=%282%2F3%29pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F3%29pi ALIGN=MIDDLE  ALT=\"%282%2F3%29pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "NOTE: I leave it to the student to determine that the radius of the semicircle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sqrt%283%29%2F3 ALIGN=MIDDLE  ALT=\"2sqrt%283%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so that the student gets to work at least part of the problem by himself.<br><BR>\n" );
document.write( "One way to find the radius is to let the radius (side length of each of the equilateral triangles) be 2x.  Then use 30-60-90 right triangles BFK and FCG to find the lengths of BF and CF in terms of x; then solve for x knowing that BF+FC=2.<br><BR>\n" );
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