document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1111020>Question 1111020</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A circle Center O touches all the sides of the regular octagon <BR>\n" );
document.write( "abcdefgh shaded in the diagram the side of the octagon are of length 12cm<BR>\n" );
document.write( "ba and gh are extended to meet at p<BR>\n" );
document.write( "hg and ef are xtended to meet q<BR>\n" );
document.write( "a)i)show that angle bah is 135 dgree  ?<BR>\n" );
document.write( "a)ii)show that angle aph is 90 degree ?\r<BR>\n" );
document.write( "\n" );
document.write( "b)calculate <BR>\n" );
document.write( "i)the length pf ph ?<BR>\n" );
document.write( "ii))the length of pq ?<BR>\n" );
document.write( "iii)the area of trangle aph ?<BR>\n" );
document.write( "iv)the area of the octangon ? \r<BR>\n" );
document.write( "\n" );
document.write( "c)calculate <BR>\n" );
document.write( "i)the radius of the circle ?<BR>\n" );
document.write( "ii)the area of the circle as a percentage of the area of the octagon ?\r<BR>\n" );
document.write( "\n" );
document.write( "please help </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #726122 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=726122\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here is the circle, centered at O, and  the octagon,<BR>\n" );
document.write( "with sides AB, CD, EF, and GH extended in both directions<BR>\n" );
document.write( "up to the point where extension meets extension:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28320%2C320%2C-16%2C16%2C-16%2C16%2C%0D%0Ared%28circle%280%2C0%2C14.485%29%29%2C%0D%0Agreen%28triangle%280%2C0%2C0%2C-14.485%2C6%2C-14.485%29%29%2C%0D%0Agreen%28rectangle%280%2C-14.485%2C0.6%2C-13.885%29%29%2C%0D%0Agreen%28triangle%28-6%2C-14.485%2C-14.485%2C-6%2C0%2C0%29%29%2C%0D%0Agreen%28triangle%28-6%2C14.485%2C-14.485%2C6%2C0%2C0%29%29%2C%0D%0Agreen%28triangle%286%2C14.485%2C14.485%2C6%2C0%2C0%29%29%2C%0D%0Agreen%28triangle%286%2C-14.485%2C14.485%2C-6%2C0%2C0%29%29%2C%0D%0Agreen%28rectangle%28-14.485%2C-14.485%2C14.485%2C14.485%29%29%2C%0D%0Atriangle%28-6%2C-14.485%2C-14.485%2C-14.485%2C-14.485%2C-6%29%2C%0D%0Atriangle%28-6%2C14.485%2C-14.485%2C14.485%2C-14.485%2C6%29%2C%0D%0Aline%28-6%2C14.485%2C6%2C14.485%29%2Cline%28-6%2C-14.485%2C6%2C-14.485%29%2C%0D%0Aline%28-14.485%2C-6%2C-14.485%2C6%29%2Cline%2814.485%2C-6%2C14.485%2C6%29%2C%0D%0Aline%28-14.485%2C-6%2C-6%2C-14.485%29%2Cline%28-14.485%2C6%2C-6%2C14.485%29%2C%0D%0Aline%2814.485%2C-6%2C6%2C-14.485%29%2Cline%2814.485%2C6%2C6%2C14.485%29%2C%0D%0Alocate%28-6.3%2C-14.5%2CA%29%2Clocate%285.7%2C-14.5%2CB%29%2C%0D%0Alocate%28-6.3%2C16%2CF%29%2Clocate%285.7%2C16%2CE%29%2C%0D%0Alocate%28-15.7%2C-5.3%2CH%29%2Clocate%28-15.7%2C6.7%2CG%29%2C%0D%0Alocate%2814.7%2C-5.3%2CC%29%2Clocate%2814.7%2C6.7%2CD%29%2C%0D%0Alocate%280.6%2C0.7%2Cred%28O%29%29%2Clocate%28-0.3%2C-14.5%2Cred%28T%29%29%2C%0D%0Alocate%28-15.7%2C-13.6%2CP%29%2Clocate%28-15.7%2C15.5%2CQ%29%2C%0D%0Alocate%2814.7%2C15.5%2Cgreen%28R%29%29%2Clocate%2814.7%2C-13.6%2Cgreen%28S%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "a) Angle BAH is one of the internal angles of the octagon,<BR>\n" );
document.write( "and it is the supplement of angle PAH,<BR>\n" );
document.write( "which is one of the external angles of the octagon.<BR>\n" );
document.write( "There is a formula about the internal angles of convex polygons,<BR>\n" );
document.write( "but I find external angle measures easier to calculate.<BR>\n" );
document.write( "The external angle of a convex polygon,<BR>\n" );
document.write( "is the change of direction as you \"turn the corner\"<BR>\n" );
document.write( "around one of the vertices of the polygon.<BR>\n" );
document.write( "For example, as you you goo from B to A,<BR>\n" );
document.write( "and turn around A to head to H,<BR>\n" );
document.write( "your change of direction is angle PAH.<BR>\n" );
document.write( "As you go a whole turn around any convex polygon,<BR>\n" );
document.write( "your changes of direction add up to a whole turn, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so its stands to reason that for a convex polygon<BR>\n" );
document.write( "the sum of the measures of the external angles is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In the case of a regular octagon, like ABCDEFGH,<BR>\n" );
document.write( "all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > angles external have the same measure,<BR>\n" );
document.write( "so angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PAH=AHP=360%5Eo%2F8=45%5Eo ALIGN=MIDDLE  ALT=\"PAH=AHP=360%5Eo%2F8=45%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Angle BAH is supplementary to PAH, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BAH=180%5Eo-PAH=180%5Eo-35%5Eo=highlight%28135%5Eo%29 ALIGN=MIDDLE  ALT=\"BAH=180%5Eo-PAH=180%5Eo-35%5Eo=highlight%28135%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Also, in triangle PAH, as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PAH=AHP=45%5Eo ALIGN=MIDDLE  ALT=\"PAH=AHP=45%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the measure of the other angle, APH is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180%5Eo-%28PAH%2BAHP%29=180%5Eo-%2845%5Eo%2B45%5Eo%29=180%5Eo-90%5Eo=highlight%2890%5Eo%29 ALIGN=MIDDLE  ALT=\"180%5Eo-%28PAH%2BAHP%29=180%5Eo-%2845%5Eo%2B45%5Eo%29=180%5Eo-90%5Eo=highlight%2890%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "b) Triangle PAH is a right triangle with two congruent <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;\" > anglem<BR>\n" );
document.write( "and that means that the opposite sides are also congruent,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AP=PH ALIGN=MIDDLE  ALT=\"AP=PH\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That triangle is an isosceles right triangle,<BR>\n" );
document.write( "so, according to the Pythagorean theorem<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AP%5E2%2BPH%5E2=AH%5E2 ALIGN=MIDDLE  ALT=\"AP%5E2%2BPH%5E2=AH%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AP=PH ALIGN=MIDDLE  ALT=\"AP=PH\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AH=12cm ALIGN=MIDDLE  ALT=\"AH=12cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2PH%5E2=%2812cm%29%5E2 ALIGN=MIDDLE  ALT=\"2PH%5E2=%2812cm%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2PH%5E2=144cm%5E2 ALIGN=MIDDLE  ALT=\"2PH%5E2=144cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PH%5E2=144cm%5E2%2F2 ALIGN=MIDDLE  ALT=\"PH%5E2=144cm%5E2%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PH%5E2=72cm%5E2 ALIGN=MIDDLE  ALT=\"PH%5E2=72cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PH=sqrt%2872cm%5E2%29=highlight%286sqrt%282%29%29 ALIGN=MIDDLE  ALT=\"PH=sqrt%2872cm%5E2%29=highlight%286sqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm ALIGN=MIDDLE  ALT=\"cm\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "As ABCDEFGH is a regular octagon,<BR>\n" );
document.write( "everything already prove/calculated about triangle PAH<BR>\n" );
document.write( "is also true about triangles GQF, ERD , and CSB.<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=GQ=PH=6sqrt%282%29cm ALIGN=MIDDLE  ALT=\"GQ=PH=6sqrt%282%29cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=HG=12cm ALIGN=MIDDLE  ALT=\"HG=12cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ=PH%2BHG%2BGQ%2B%286sqrt%282%29%2B12%2B6sqrt%282%29%29cm=highlight%2812%2B12sqrt%282%29%29 ALIGN=MIDDLE  ALT=\"PQ=PH%2BHG%2BGQ%2B%286sqrt%282%29%2B12%2B6sqrt%282%29%29cm=highlight%2812%2B12sqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm ALIGN=MIDDLE  ALT=\"cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Of course, the other sides of quadrilateral PQRS also measure<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2812%2B12sqrt%282%29%29cm ALIGN=MIDDLE  ALT=\"%2812%2B12sqrt%282%29%29cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and PQRS is a square.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%28APH%29=AP%2APH%2F2=%286sqrt%282%29cm%29%2A%286sqrt%282%29cm%29%2F2=highlight%2836cm%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and each of the areas of GQF, ERD , and CSB are also <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36cm%5E2 ALIGN=MIDDLE  ALT=\"36cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%28ABCDEFGH%29=area%28PQRS%29-%28area%28PAH%29%2Barea%28GQF%29%2Barea%28ERD%29%2Barea%28CSB%29%29=area%28PQRS%29-4%2Aarea%28PAH%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%28PQRS%29=%28%2812%2B12sqrt%282%29%29cm%29%5E2=%28144%2B288%2B288sqrt%282%29%29cm%5E2 ALIGN=MIDDLE  ALT=\"area%28PQRS%29=%28%2812%2B12sqrt%282%29%29cm%29%5E2=%28144%2B288%2B288sqrt%282%29%29cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%28ABCDEFGH%29=%28144%2B288%2B288sqrt%282%29%29cm%5E2-4%2A36cm%5E2=%28144%2B288%2B288sqrt%282%29%29cm%5E2-144cm%5E2=highlight%28288%2B288sqrt%282%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E2 ALIGN=MIDDLE  ALT=\"cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "c) For a regular polygon, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=perimeter%2Aapothem%2F2 ALIGN=MIDDLE  ALT=\"area=perimeter%2Aapothem%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The apothem of ABCDEFGH is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OT ALIGN=MIDDLE  ALT=\"OT\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and is the radius of the circle.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%28ABCDEFGH%29=%28288%2B288sqrt%282%29%29cm%5E2 ALIGN=MIDDLE  ALT=\"area%28ABCDEFGH%29=%28288%2B288sqrt%282%29%29cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=perimeter%28ABCDEFGH%29=8%2A%2812cm%29=96cm ALIGN=MIDDLE  ALT=\"perimeter%28ABCDEFGH%29=8%2A%2812cm%29=96cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28288%2B288sqrt%282%29%29cm%5E2=%2896cm%29%28OT%29%2F2 ALIGN=MIDDLE  ALT=\"%28288%2B288sqrt%282%29%29cm%5E2=%2896cm%29%28OT%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Solving for OT, we find the radius of the circle as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OT=2%2A%28288%2B288sqrt%282%29%29%2F96 ALIGN=MIDDLE  ALT=\"OT=2%2A%28288%2B288sqrt%282%29%29%2F96\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm=highlight%286%2B6sqrt%282%29%29 ALIGN=MIDDLE  ALT=\"cm=highlight%286%2B6sqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm ALIGN=MIDDLE  ALT=\"cm\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The area of the circle is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2Aradius%5E2=pi%286%2B6sqrt%282%29%29%5E2 ALIGN=MIDDLE  ALT=\"pi%2Aradius%5E2=pi%286%2B6sqrt%282%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E2=pi%2836%2B72%2B72sqrt%282%29%29cm%5E2=pi%28108%2B72sqrt%282%29%29cm%5E2 ALIGN=MIDDLE  ALT=\"cm%5E2=pi%2836%2B72%2B72sqrt%282%29%29cm%5E2=pi%28108%2B72sqrt%282%29%29cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The ratio of that area to the area of the octagon is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%28108%2B72sqrt%282%29%29%2F%28288%2B288sqrt%282%29%29=about0.948 ALIGN=MIDDLE  ALT=\"pi%28108%2B72sqrt%282%29%29%2F%28288%2B288sqrt%282%29%29=about0.948\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "As <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.948=94.8%2F100=%2294.8%25%22 ALIGN=MIDDLE  ALT=\"0.948=94.8%2F100=%2294.8%25%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "The area of the circle is approximately <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28%2294.8%25%22%29 ALIGN=MIDDLE  ALT=\"highlight%28%2294.8%25%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the area of the octagon. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );