document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=947419>Question 947419</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Quadrilateral ABCD is a rhombus. Find X, Y, z, and the perimeter of ABCD\r<BR>\n" );
document.write( "\n" );
document.write( "measure of angle BCD=106 degrees <BR>\n" );
document.write( "measure of angle ADB= (4y+5) degrees<BR>\n" );
document.write( "measure of angle BAC= (6x-7) degrees<BR>\n" );
document.write( "EC=6 <BR>\n" );
document.write( "ED=5z-7<BR>\n" );
document.write( "BE= 3z-1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #578215 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=578215\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It is not absolutely necessary to sketch the rhombus,<BR>\n" );
document.write( "but it would help visualize the relations between angles.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28240%2C300%2C-1.2%2C1.2%2C-1.5%2C1.5%2C%0D%0Aline%28-1%2C0%2C0%2C1.327%29%2Cline%281%2C0%2C0%2C1.327%29%2C%0D%0Aline%28-1%2C0%2C0%2C-1.327%29%2Cline%281%2C0%2C0%2C-1.327%29%2C%0D%0Agreen%28line%28-1%2C0%2C1%2C0%29%29%2Cgreen%28line%280%2C-1.327%2C0%2C1.327%29%29%2C%0D%0Alocate%28-1.1%2C0%2CA%29%2Clocate%281%2C0%2CC%29%2C%0D%0Alocate%280.05%2C1.4%2CB%29%2Clocate%280.05%2C-1.3%2CD%29%2C%0D%0Ared%28arc%281%2C0%2C1%2C1%2C127%2C233%29%29%2C%0D%0Alocate%280.55%2C0.03%2Cred%28106%5Eo%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Angles BCD and BAD are opposite angles, so they have the same measure:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BAD=DCD=106%5Eo ALIGN=MIDDLE  ALT=\"BAD=DCD=106%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Angle BAC (with side BA and diagonal AC) is half of angle BAD,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BAC=BAD%2F2=BCD%2F2=106%5Eo%2F2=53%5Eo ALIGN=MIDDLE  ALT=\"BAC=BAD%2F2=BCD%2F2=106%5Eo%2F2=53%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Without the drawing, I can tell that<BR>\n" );
document.write( "ADB is one of the base angles of isosceles triangle ADB<BR>\n" );
document.write( "(with diagonal BD as its base,<BR>\n" );
document.write( "and angle BAD as its vertex angle),<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ADB=%28180%5Eo-BAD%29%2F2=%28180%5Eo-106%5Eo%29%2F2=74%5Eo%2F2=37%5Eo ALIGN=MIDDLE  ALT=\"ADB=%28180%5Eo-BAD%29%2F2=%28180%5Eo-106%5Eo%29%2F2=74%5Eo%2F2=37%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "With the drawing, you can tell that<BR>\n" );
document.write( "since the diagonals of a rhombus are perpendicular to each other,<BR>\n" );
document.write( "angles ADB and BAC are acute angles in a right triangle,<BR>\n" );
document.write( "and therefore complementary, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ADB=90%5Eo-BAC=90%5Eo-53%5Eo=37%5Eo ALIGN=MIDDLE  ALT=\"ADB=90%5Eo-BAC=90%5Eo-53%5Eo=37%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "With the angle measures in hand, the equations to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y%2B5=37 ALIGN=MIDDLE  ALT=\"4y%2B5=37\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x-7=53 ALIGN=MIDDLE  ALT=\"6x-7=53\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y%2B5=37 ALIGN=MIDDLE  ALT=\"4y%2B5=37\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y=37-5 ALIGN=MIDDLE  ALT=\"4y=37-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y=32 ALIGN=MIDDLE  ALT=\"4y=32\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=32%2F4 ALIGN=MIDDLE  ALT=\"y=32%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=8%29 ALIGN=MIDDLE  ALT=\"highlight%28y=8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x-7=53 ALIGN=MIDDLE  ALT=\"6x-7=53\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x=53%2B7 ALIGN=MIDDLE  ALT=\"6x=53%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x=60 ALIGN=MIDDLE  ALT=\"6x=60\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=60%2F6 ALIGN=MIDDLE  ALT=\"x=60%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=10%29 ALIGN=MIDDLE  ALT=\"highlight%28x=10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "You are not telling me what/where point E is,<BR>\n" );
document.write( "but I am guessing that it is the intersection of the diagonals of the rhombus.<BR>\n" );
document.write( "Since B and D are opposite vertices, BD is a diagonal,<BR>\n" );
document.write( "and if E is the intersection of both diagonals,<BR>\n" );
document.write( "then E is the midpoint of both diagonals and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ED=BE ALIGN=MIDDLE  ALT=\"ED=BE\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That gives us the equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5z-7=3z-1 ALIGN=MIDDLE  ALT=\"5z-7=3z-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5z-3z=7-1 ALIGN=MIDDLE  ALT=\"5z-3z=7-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2z=6 ALIGN=MIDDLE  ALT=\"2z=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=6%2F3 ALIGN=MIDDLE  ALT=\"z=6%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28z=3%29 ALIGN=MIDDLE  ALT=\"highlight%28z=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That makes<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ED=5%2A3-7=15-7=8 ALIGN=MIDDLE  ALT=\"ED=5%2A3-7=15-7=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BE=3%2A3-1=9-1=8 ALIGN=MIDDLE  ALT=\"BE=3%2A3-1=9-1=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which verifies.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Most importantly, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BE=8 ALIGN=MIDDLE  ALT=\"BE=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=EC=6 ALIGN=MIDDLE  ALT=\"EC=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "are the legs of right triangle BEC, so<BR>\n" );
document.write( "the Pythagorean theorem says that the length of the hypotenuse is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC=sqrt%288%5E2%2B6%5E2%29=sqrt%2864%2B36%29=sqrt%28100%29=10 ALIGN=MIDDLE  ALT=\"BC=sqrt%288%5E2%2B6%5E2%29=sqrt%2864%2B36%29=sqrt%28100%29=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That makes the length of each side of ABCD <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in whatever units),<BR>\n" );
document.write( "and the perimeter of ABCD is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A10=highlight%2840%29 ALIGN=MIDDLE  ALT=\"4%2A10=highlight%2840%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "The teacher proposing this problem used a 2X scaled up version of the very popular right triangle with side lengths 3, 4, and 5 to design the rhombus.<BR>\n" );
document.write( "That would make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28BAC%29=8%2F6=4%2F3=1.333%28rounded%29 ALIGN=MIDDLE  ALT=\"tan%28BAC%29=8%2F6=4%2F3=1.333%28rounded%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "However, the solution to the problem involves <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BAC=53%5Eo ALIGN=MIDDLE  ALT=\"BAC=53%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2853%5Eo%29=1.327%28rounded%29 ALIGN=MIDDLE  ALT=\"tan%2853%5Eo%29=1.327%28rounded%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which is not quite equal,<BR>\n" );
document.write( "but very close to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F3=1.333%28rounded%29 ALIGN=MIDDLE  ALT=\"4%2F3=1.333%28rounded%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Using the fact that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%2853%5Eo%29=0.7986%28rounded%29 ALIGN=MIDDLE  ALT=\"sin%2853%5Eo%29=0.7986%28rounded%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "you would not have needed the length of EC,<BR>\n" );
document.write( "and could have calculated <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC=approximately ALIGN=MIDDLE  ALT=\"BC=approximately\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2F0.7986=10.02 ALIGN=MIDDLE  ALT=\"8%2F0.7986=10.02\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That tells me that you were obviously not expected to use trigonometric functions, but the problem designer tried to make it believable by choosing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=106%5Eo ALIGN=MIDDLE  ALT=\"106%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > for angle BCD. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );