document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1091564>Question 1091564</A>: <I><SPAN STYLE=\"word-break: break-all;\"> XZ is a chord which is 12cm long.If the perpendicular distance from the midpoint of the chord to point Y on the circumference of the circle is 4cm,calculate,correct to 1d.p,the perimeter of sector OXYZ.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #705988 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5367)</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=705988\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here is the circle, with the chord XZ, and point Y.<BR>\n" );
document.write( "The chord's midpoint, M, and the circle circle center, O, are also labeled.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-7.5%2C7.5%2C-7.5%2C7.5%2C%0D%0Acircle%280%2C0%2C6.5%29%2Cline%280%2C6.5%2C0%2C2.5%29%2C%0D%0Aline%28-6%2C2.5%2C6%2C2.5%29%2Ccircle%280%2C0%2C0.1%29%2C%0D%0Arectangle%280%2C2.5%2C0.4%2C2.9%29%2Clocate%280.2%2C-0.1%2CO%29%2C%0D%0Alocate%280.1%2C4.8%2C4cm%29%2C%0D%0Alocate%282.5%2C2.5%2C6cm%29%2Clocate%28-3.5%2C2.5%2C6cm%29%2C%0D%0Alocate%28-6.5%2C3%2CX%29%2Clocate%286.2%2C3%2CZ%29%2C%0D%0Alocate%28-0.6%2C3.2%2CM%29%2Clocate%28-0.2%2C7.2%2CY%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Here is the perimeter of sector OXYZ, colored in red.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-7.5%2C7.5%2C-7.5%2C7.5%2C%0D%0Aarc%280%2C0%2C13%2C13%2C337.38%2C202.62%29%2Ccircle%280%2C0%2C0.1%29%2C%0D%0Atriangle%280%2C0%2C6%2C2.5%2C-6%2C2.5%29%2Cline%280%2C6.5%2C0%2C2.5%29%2C%0D%0Ared%28arc%280%2C0%2C13.1%2C13.1%2C202.62%2C337.38%29%29%2C%0D%0Ared%28arc%280%2C0%2C13%2C13%2C202.62%2C337.38%29%29%2C%0D%0Ared%28line%280%2C0%2C6%2C2.5%29%29%2Cred%28line%280%2C0%2C-6%2C2.5%29%29%2C%0D%0Alocate%280.2%2C-0.1%2CO%29%2Clocate%280.1%2C4.8%2C4cm%29%2C%0D%0Alocate%28-0.4%2C2.5%2C12cm%29%2Crectangle%280%2C2.5%2C0.4%2C2.9%29%2C%0D%0Alocate%28-6.5%2C3%2CX%29%2Clocate%286.2%2C3%2CZ%29%2C%0D%0Alocate%28-0.6%2C3.2%2CM%29%2Clocate%28-0.2%2C7.2%2CY%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We needed to draw radii OX and OZ, of course.<BR>\n" );
document.write( "While we are at it, let's also extend segment YM<BR>\n" );
document.write( "to draw full radius OY  and full diameter YP,<BR>\n" );
document.write( "and draw a few more segments (XY, YZ, and ZP).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-7.5%2C7.5%2C-7.5%2C7.5%2C%0D%0Acircle%280%2C0%2C6.5%29%2Ctriangle%280%2C6.5%2C6%2C2.5%2C-6%2C2.5%29%2C%0D%0Atriangle%280%2C0%2C-6%2C2.5%2C6%2C2.5%29%2Ccircle%280%2C0%2C0.1%29%2C%0D%0Arectangle%280%2C2.5%2C0.4%2C2.9%29%2Ctriangle%280%2C6.5%2C0%2C-6.5%2C6%2C2.5%29%2C%0D%0Alocate%280.2%2C-0.1%2CO%29%2Clocate%280.1%2C4.8%2C4cm%29%2C%0D%0Alocate%282.5%2C2.5%2C6cm%29%2Clocate%28-3.5%2C2.5%2C6cm%29%2C%0D%0Alocate%28-6.5%2C3%2CX%29%2Clocate%286.2%2C3%2CZ%29%2Clocate%28-0.2%2C-6.5%2CP%29%2C%0D%0Alocate%28-0.6%2C3.2%2CM%29%2Clocate%28-0.2%2C7.2%2CY%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We have for radii, and will call their length <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > cm.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=OX=OZ=OY=OP ALIGN=MIDDLE  ALT=\"r=OX=OZ=OY=OP\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We have four obvious right angles at point M.<BR>\n" );
document.write( "We also have right angle YZP, that is not so obvious,<BR>\n" );
document.write( "but is a right angle, because it intercepts diameter YP.<BR>\n" );
document.write( "That makes for a whole bunch of right triangles,<BR>\n" );
document.write( "but if we look look at YZP, YMZ, and ZMP,<BR>\n" );
document.write( "we see that they are similar right triangles.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-7.5%2C7.5%2C-7.5%2C7.5%2C%0D%0Acircle%280%2C0%2C6.5%29%2Ctriangle%280%2C6.5%2C6%2C2.5%2C-6%2C2.5%29%2C%0D%0Atriangle%280%2C0%2C-6%2C2.5%2C6%2C2.5%29%2Ccircle%280%2C0%2C0.1%29%2C%0D%0Arectangle%280%2C2.5%2C0.4%2C2.9%29%2Ctriangle%280%2C6.5%2C0%2C-6.5%2C6%2C2.5%29%2C%0D%0Alocate%280.2%2C-0.1%2CO%29%2Clocate%280.1%2C4.8%2C4cm%29%2C%0D%0Alocate%282.5%2C2.5%2C6cm%29%2Clocate%28-3.5%2C2.5%2C6cm%29%2C%0D%0Alocate%28-6.5%2C3%2CX%29%2Clocate%286.2%2C3%2CZ%29%2Clocate%28-0.2%2C-6.5%2CP%29%2C%0D%0Alocate%28-0.6%2C3.2%2CM%29%2Clocate%28-0.2%2C7.2%2CY%29%2C%0D%0Ared%28arc%280%2C6.5%2C2%2C2%2C33.69%2C90%29%29%2Cgreen%28arc%286%2C2.5%2C2%2C2%2C180%2C213.69%29%29%2C%0D%0Agreen%28arc%286%2C2.5%2C2.5%2C2.5%2C180%2C213.69%29%29%2C%0D%0Agreen%28arc%280%2C-6.5%2C2%2C2%2C270%2C303.69%29%29%2Cgreen%28arc%280%2C-6.5%2C2.5%2C2.5%2C270%2C303.69%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Right triangles ZMP and YMZ are similar,<BR>\n" );
document.write( "so their long leg ratios and short leg ratios<BR>\n" );
document.write( "(or long leg to short leg ratios) are the same.<BR>\n" );
document.write( "Either ratio equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PM%2FZM=ZM%2FYM ALIGN=MIDDLE  ALT=\"PM%2FZM=ZM%2FYM\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282r-4%29%2F6=6%2F4 ALIGN=MIDDLE  ALT=\"%282r-4%29%2F6=6%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%282r-4%29=6%2A6 ALIGN=MIDDLE  ALT=\"4%282r-4%29=6%2A6\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8r-16=36 ALIGN=MIDDLE  ALT=\"8r-16=36\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8r=52 ALIGN=MIDDLE  ALT=\"8r=52\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=52%2F8 ALIGN=MIDDLE  ALT=\"r=52%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=6.5 ALIGN=MIDDLE  ALT=\"r=6.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we just need to find the length of arc XYZ,<BR>\n" );
document.write( "which by definition is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > times the measure in radians of<BR>\n" );
document.write( "The angle XOZ that contains Y.<BR>\n" );
document.write( "We can figure out the measure of the acute angles marked in green in those right triangles:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28YZX%29=4cm%2F%226+cm%22=2%2F3 ALIGN=MIDDLE  ALT=\"tan%28YZX%29=4cm%2F%226+cm%22=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That corresponds to approximately <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.588003 ALIGN=MIDDLE  ALT=\"0.588003\"   DISABLED_style= \"max-width:90%; height:auto;\" > radians.<BR>\n" );
document.write( "That is the measure of YZX and YPZ, that are inscribed in the circle.<BR>\n" );
document.write( "The central angles intercepting the same arcs,<BR>\n" );
document.write( "acute angles YOX and YOZ, measure twice as much.<BR>\n" );
document.write( "The measure of their sum, the angle XOZ containing point Y, is four times as much,<BR>\n" );
document.write( "about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A0.588=2.352 ALIGN=MIDDLE  ALT=\"4%2A0.588=2.352\"   DISABLED_style= \"max-width:90%; height:auto;\" > radians.<BR>\n" );
document.write( "So, the length, in cm, of arc XYZ is about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6.5%2A2.352=15.288 ALIGN=MIDDLE  ALT=\"6.5%2A2.352=15.288\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Now we can calculate the perimeter of sector OXYZ,<BR>\n" );
document.write( "by adding the length of arc XYZ,<BR>\n" );
document.write( "plus the length of radii OX and OZ :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15.288cm%2B6.5cm%2B6.5cm-288.288cm ALIGN=MIDDLE  ALT=\"15.288cm%2B6.5cm%2B6.5cm-288.288cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Rounded to 1 decimal place, it is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2828.9cm%29 ALIGN=MIDDLE  ALT=\"highlight%2828.9cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-7.5%2C7.5%2C-7.5%2C7.5%2C%0D%0Aarc%280%2C0%2C13%2C13%2C337.38%2C202.62%29%2Ccircle%280%2C0%2C0.1%29%2C%0D%0Atriangle%280%2C0%2C6%2C2.5%2C-6%2C2.5%29%2Cline%280%2C6.5%2C0%2C2.5%29%2C%0D%0Ared%28arc%280%2C0%2C13.1%2C13.1%2C202.62%2C337.38%29%29%2C%0D%0Ared%28arc%280%2C0%2C13%2C13%2C202.62%2C337.38%29%29%2C%0D%0Ared%28line%280%2C0%2C6%2C2.5%29%29%2Cred%28line%280%2C0%2C-6%2C2.5%29%29%2C%0D%0Alocate%280.2%2C-0.1%2CO%29%2Clocate%280.1%2C4.8%2C4cm%29%2C%0D%0Alocate%28-0.4%2C2.5%2C12cm%29%2Crectangle%280%2C2.5%2C0.4%2C2.9%29%2C%0D%0Alocate%28-6.5%2C3%2CX%29%2Clocate%286.2%2C3%2CZ%29%2C%0D%0Alocate%28-0.6%2C3.2%2CM%29%2Clocate%28-0.2%2C7.2%2CY%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );