document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=732663>Question 732663</A>: <I><SPAN STYLE=\"word-break: break-all;\"> AOB and COD are two perpendicular diameters of a circle with radius 4 feet. With center A and radius AB an arc is drawn from B to meet AC extended at P, and with center B and radius BA an arc is drawn from A to meet BC extended at Q. With center C the arc PQ is drawn. DC extended meets this arc at R. Find DR and the perimeter of ADBPRQ. \r<BR>\n" );
document.write( "\n" );
document.write( "So far, line segment AO, OC, OB, and OD are all 4 feet. Triangle ACO and OCB are 45-45-90. Line segments AC and CB are 3 root 2. Angles QCR and RCP are each 45 degrees, making arc QP 90 degrees. I really don't know where to go from here. Can someone please help? I have been stuck on this for days! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #448047 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=448047\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C400%2C-4.5%2C4.5%2C-5%2C7%2C%0D%0Acircle%280%2C0%2C4%29%2C%0D%0Aline%284%2C0%2C-4%2C0%29%2Cline%280%2C7%2C0%2C-4%29%2C%0D%0Ared%28line%28-4%2C0%2C0%2C-4%29%29%2Cred%28line%280%2C-4%2C4%2C0%29%29%2C%0D%0Acircle%284%2C0%2C8%29%2Ccircle%28-4%2C0%2C8%29%2C%0D%0Agreen%28circle%280%2C4%2C2.343%29%29%2C%0D%0Aline%28-4%2C0%2C4%2C8%29%2Cline%284%2C0%2C-4%2C8%29%2C%0D%0Ared%28line%28-4%2C0%2C-1.657%2C5.657%29%29%2Cred%28line%284%2C0%2C1.657%2C5.657%29%29%2C%0D%0Ared%28line%28-1.657%2C5.657%2C0%2C6.343%29%29%2Cred%28line%280%2C6.343%2C1.657%2C5.657%29%29%2C%0D%0Alocate%28-0.1%2C-4.1%2CD%29%2Clocate%28-4.3%2C0.2%2CA%29%2Clocate%284.1%2C0.2%2CB%29%2C%0D%0Alocate%280.1%2C-0.1%2CO%29%2Clocate%280.1%2C3.7%2CC%29%2C%0D%0Alocate%281.8%2C5.9%2CP%29%2Clocate%28-2%2C5.9%2CQ%29%2Clocate%280.1%2C6.2%2CR%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > I did not know how to draw just the arc PQR, so I had to draw the whole green circle.<BR>\n" );
document.write( "The sides of the square ADBC are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4sqrt%282%29=about ALIGN=MIDDLE  ALT=\"4sqrt%282%29=about\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5.657feet ALIGN=MIDDLE  ALT=\"5.657feet\"   DISABLED_style= \"max-width:90%; height:auto;\" > in length.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CD=AB=AP=BQ=8 ALIGN=MIDDLE  ALT=\"CD=AB=AP=BQ=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > because they are radii of the circles containing the arcs BP and AQ<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CR=CP=CQ=8-4sqrt%282%29 ALIGN=MIDDLE  ALT=\"CR=CP=CQ=8-4sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > all radii of my green circle containing arc PRQ<BR>\n" );
document.write( " <BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DR=CD%2BCR=8%2B8-4sqrt%282%29=16-4sqrt%282%29=about ALIGN=MIDDLE  ALT=\"DR=CD%2BCR=8%2B8-4sqrt%282%29=16-4sqrt%282%29=about\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10.343feet ALIGN=MIDDLE  ALT=\"10.343feet\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "BPA and QAB are isosceles triangles with a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=45%5Eo ALIGN=MIDDLE  ALT=\"45%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > vertex angle and legs measuring 8 feet.<BR>\n" );
document.write( "Based on law of cosines or  using the fact that BPC and AQC are right triangles, we can calculate that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BP%5E2=AQ%5E2=64%282-sqrt%282%29%29=about ALIGN=MIDDLE  ALT=\"BP%5E2=AQ%5E2=64%282-sqrt%282%29%29=about\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=37.49feet%5E2 ALIGN=MIDDLE  ALT=\"37.49feet%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The approximate length would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BP=AQ=sqrt%2864%282-sqrt%282%29%29%29=8sqrt%282-sqrt%282%29%29=about ALIGN=MIDDLE  ALT=\"BP=AQ=sqrt%2864%282-sqrt%282%29%29%29=8sqrt%282-sqrt%282%29%29=about\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6.123feet ALIGN=MIDDLE  ALT=\"6.123feet\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Otherwise we could split those triangles into two congruent right triangles with a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=22.5%5Eo ALIGN=MIDDLE  ALT=\"22.5%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > angle and 8-foot hypotenuse, and calculate the length of their short legs (in feet) as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BP%2F2=AQ%2F2=8sin%2822.5%5Eo%29=about ALIGN=MIDDLE  ALT=\"BP%2F2=AQ%2F2=8sin%2822.5%5Eo%29=about\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2A0.3827=3.0615 ALIGN=MIDDLE  ALT=\"8%2A0.3827=3.0615\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Either way the ratio of base to leg length in those isosceles triangles is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282-sqrt%282%29%29=about ALIGN=MIDDLE  ALT=\"sqrt%282-sqrt%282%29%29=about\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.7654 ALIGN=MIDDLE  ALT=\"0.7654\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "PRC and RQC are also isosceles triangles with a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=45%5Eo ALIGN=MIDDLE  ALT=\"45%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > vertex angle, so they are similar to BPC and AQC.<BR>\n" );
document.write( "We knew that the length of their legs (in feet) were<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CR=CP=CQ=8-4sqrt%282%29 ALIGN=MIDDLE  ALT=\"CR=CP=CQ=8-4sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and multiplying that times the ratio found above for the similar triangles we can find the length of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PR=RQ ALIGN=MIDDLE  ALT=\"PR=RQ\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Giving up on accurate value expressions,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CR=CP=CQ=8-4sqrt%282%29=about ALIGN=MIDDLE  ALT=\"CR=CP=CQ=8-4sqrt%282%29=about\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.343 ALIGN=MIDDLE  ALT=\"2.343\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PR%2BPQ=about ALIGN=MIDDLE  ALT=\"PR%2BPQ=about\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.793feet ALIGN=MIDDLE  ALT=\"1.793feet\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Now we can calculate the perimeter of ADBPRQ as the approximate value (in feet) of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5.657%2B5.657%2B6.123%2B6.123%2B1.793%2B1.793=27.146 ALIGN=MIDDLE  ALT=\"5.657%2B5.657%2B6.123%2B6.123%2B1.793%2B1.793=27.146\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );