document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1056011>Question 1056011</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The coordinates of the vertices of quadrilateral NOVA are (-1,4), (4,7), (7,2) and (2,-1) respectively. Classify quadrilateral NOVA (square, rectangle or rhombus) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #671294 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=671294\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The wording suggests that square, rectangle or rhombus are the only choices,<BR>\n" );
document.write( "as if they are telling you that it is not any quadrilateral,<BR>\n" );
document.write( "but one of those 3 kinds of parallelograms.<BR>\n" );
document.write( "Was that what was really meant?<BR>\n" );
document.write( "If you never learned about slopes,<BR>\n" );
document.write( "that must be what was meant.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Calculating <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28lengths%29 ALIGN=MIDDLE  ALT=\"highlight%28lengths%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > you could verify<BR>\n" );
document.write( "if a quadrilateral is a rectangle, a square or a rhombus.<BR>\n" );
document.write( "In a square or a rhombus, all 4 sides have the same length.<BR>\n" );
document.write( "In a square of a rectangle, both diagonals have the same length.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28slopes%29 ALIGN=MIDDLE  ALT=\"highlight%28slopes%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the segments (sides or diagonals) would tell you if they are parallel or perpendicular, or neither,<BR>\n" );
document.write( "and that would allow you to know if a quadrilateral is a parallelogram,<BR>\n" );
document.write( "and in that case it would tell you what kind.<BR>\n" );
document.write( "In a parallelogram, opposite sides are parallel;<BR>\n" );
document.write( "in a rectangle, adjacent sides are perpendicular to each other,<BR>\n" );
document.write( "and in a rhombus,the diagonals are prependicular to each other.<BR>\n" );
document.write( " <BR>\n" );
document.write( "CALCULATING LENGTHS ONLY:<BR>\n" );
document.write( "The \"formula\" for calculating the length of a line segment,<BR>\n" );
document.write( "is often called \"the distance formula\" ,<BR>\n" );
document.write( "because it is the distance between the end points of the segment.<BR>\n" );
document.write( "It is not really a formula to be memorized,<BR>\n" );
document.write( "because it is just the application of the Pythagorean theorem<BR>\n" );
document.write( "to the right triangles formed when you draw the segment between the points,<BR>\n" );
document.write( "and through each point draw lines parallel to the x- and y-axes.<BR>\n" );
document.write( "For segment <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=NO ALIGN=MIDDLE  ALT=\"NO\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the length could be calculated by the formula<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=NO=sqrt%28%28x%5BO%5D-x%5BN%5D%29%5E2%2B%28y%5BO%5D-y%5BN%5D%29%5E2%29 ALIGN=MIDDLE  ALT=\"NO=sqrt%28%28x%5BO%5D-x%5BN%5D%29%5E2%2B%28y%5BO%5D-y%5BN%5D%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5BO%5D-x%5BN%5D%29=-%28x%5BN%5D-x%5BO%5D%29 ALIGN=MIDDLE  ALT=\"%28x%5BO%5D-x%5BN%5D%29=-%28x%5BN%5D-x%5BO%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5BO%5D-x%5BN%5D%29%5E2=%28x%5BN%5D-x%5BO%5D%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%5BO%5D-x%5BN%5D%29%5E2=%28x%5BN%5D-x%5BO%5D%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and that means that you do not need to worry about which way you write the squared differences.<BR>\n" );
document.write( "So, the lengths of the sides are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=NO=sqrt%28%284-1%29%5E2%2B%287-4%29%5E2%29=sqrt%283%5E2%2B5%5E2%29=sqrt%289%2B25%29=sqrt%2834%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=OV=sqrt%28%284-7%29%5E2%2B%287-2%29%5E2%29=sqrt%28%28-3%29%5E2%2B5%5E2%29=sqrt%289%2B25%29=sqrt%2834%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=VA=sqrt%28%282-7%29%5E2%2B%28-1-2%29%5E2%29=sqrt%28%28-5%29%5E2%2B%28-3%29%5E2%29=sqrt%289%2B25%29=sqrt%2834%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=NA=sqrt%28%284-7%29%5E2%2B%287-2%29%5E2%29=sqrt%28%28-3%29%5E2%2B5%5E2%29=sqrt%289%2B25%29=sqrt%2834%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The lengths of the diagonals are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=NV=sqrt%28%282-4%29%5E2%2B%287-%28-1%29%29%5E2%29=sqrt%28%28-2%29%5E2%2B%287%2B1%29%5E2%29=sqrt%284%2B8%5E2%29=sqrt%284%2B64%29=sqrt%2868%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OA=sqrt%28%282-4%29%5E2%2B%28-1-7%29%5E2%29=sqrt%28%28-2%29%5E2%2B%28-8%29%5E2%29=sqrt%284%2B64%29=sqrt%2868%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Since all four sides have the same <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2834%29 ALIGN=MIDDLE  ALT=\"sqrt%2834%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > length,<BR>\n" );
document.write( "and both diagonals have the same <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2868%29 ALIGN=MIDDLE  ALT=\"sqrt%2868%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > length,<BR>\n" );
document.write( "quadrilateral <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=NOVA ALIGN=MIDDLE  ALT=\"NOVA\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a square.<BR>\n" );
document.write( "I would also call it a rectangle and a rhombus,<BR>\n" );
document.write( "because a square is a special kind of rectangle (one with all sides having the same length),<BR>\n" );
document.write( "and a special kind of rhombus (one with all angles having the same measure).\r<BR>\n" );
document.write( "\n" );
document.write( " <BR>\n" );
document.write( "CALCULATING SLOPES:<BR>\n" );
document.write( "A short way to write the definition for slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=DELTA%28y%29%2FDELTA%28x%29 ALIGN=MIDDLE  ALT=\"m=DELTA%28y%29%2FDELTA%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but that assumes you know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > means slope, and that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DELTA ALIGN=MIDDLE  ALT=\"DELTA\"   DISABLED_style= \"max-width:90%; height:auto;\" > means change (increase really).<BR>\n" );
document.write( "So, they usually make you write something more complicated, like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Slope%5BNO%5D ALIGN=MIDDLE  ALT=\"Slope%5BNO%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=or ALIGN=MIDDLE  ALT=\"or\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BNO%5D=%28y%5BO%5D-y%5BN%5D%29%2F%28x%5BO%5D-x%5BN%5D%29 ALIGN=MIDDLE  ALT=\"m%5BNO%5D=%28y%5BO%5D-y%5BN%5D%29%2F%28x%5BO%5D-x%5BN%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "There, the increase in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > as you go from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N ALIGN=MIDDLE  ALT=\"N\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=O ALIGN=MIDDLE  ALT=\"O\"   DISABLED_style= \"max-width:90%; height:auto;\" > is  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5BO%5D-y%5BN%5D=DELTA%28y%29 ALIGN=MIDDLE  ALT=\"y%5BO%5D-y%5BN%5D=DELTA%28y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "the increase in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > as you go from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N ALIGN=MIDDLE  ALT=\"N\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=O ALIGN=MIDDLE  ALT=\"O\"   DISABLED_style= \"max-width:90%; height:auto;\" > is  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5BO%5D-x%5BN%5D=DELTA%28x%29 ALIGN=MIDDLE  ALT=\"x%5BO%5D-x%5BN%5D=DELTA%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "That \"formula\" is a definition of the word slope.<BR>\n" );
document.write( "You just need to remember its meaning,<BR>\n" );
document.write( "because that is not something you can deduce by reasoning,<BR>\n" );
document.write( "but it is not really a complicated \"formula\" that you need to memorize.<BR>\n" );
document.write( "So, we use that type of \"formula\" to calculate the slopes of sides <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=NO ALIGN=MIDDLE  ALT=\"NO\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OV ALIGN=MIDDLE  ALT=\"OV\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=VA ALIGN=MIDDLE  ALT=\"VA\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AN ALIGN=MIDDLE  ALT=\"AN\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "When I fill numbers into that formula,<BR>\n" );
document.write( "I first fill the x and y of one point,<BR>\n" );
document.write( "and then I fill the coordinates of the other point,<BR>\n" );
document.write( "to avoid mistakes.<BR>\n" );
document.write( "I would first enter the coordinates for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=O ALIGN=MIDDLE  ALT=\"O\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BNO%5D=%287-%22+%22%29%2F%284-%22+%22%29 ALIGN=MIDDLE  ALT=\"m%5BNO%5D=%287-%22+%22%29%2F%284-%22+%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and then I would fill the blanks with the coordinates for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N ALIGN=MIDDLE  ALT=\"N\"   DISABLED_style= \"max-width:90%; height:auto;\" > , to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BNO%5D=%287-4%29%2F%284-%28-1%29%29 ALIGN=MIDDLE  ALT=\"m%5BNO%5D=%287-4%29%2F%284-%28-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which is what my teacher would see. Then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BNO%5D=3%2F%284%2B1%29 ALIGN=MIDDLE  ALT=\"m%5BNO%5D=3%2F%284%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BNO%5D=3%2F5 ALIGN=MIDDLE  ALT=\"m%5BNO%5D=3%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The other sides' slopes can be calculated the same way:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BOV%5D=%282-7%29%2F%287-4%29=%28-5%29%2F3=-5%2F3 ALIGN=MIDDLE  ALT=\"m%5BOV%5D=%282-7%29%2F%287-4%29=%28-5%29%2F3=-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BVA%5D=%28-1-2%29%2F%282-7%29=%28-3%29%2F%28-5%29=3%2F5 ALIGN=MIDDLE  ALT=\"m%5BVA%5D=%28-1-2%29%2F%282-7%29=%28-3%29%2F%28-5%29=3%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BAN%5D=%28-1-4%29%2F%282-%28-1%29%29=%28-5%29%2F%282%2B1%29=%28-5%29%2F3=-5%2F3 ALIGN=MIDDLE  ALT=\"m%5BAN%5D=%28-1-4%29%2F%282-%28-1%29%29=%28-5%29%2F%282%2B1%29=%28-5%29%2F3=-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BNO%5D=m%5BVA%5D=3%2F5 ALIGN=MIDDLE  ALT=\"m%5BNO%5D=m%5BVA%5D=3%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > , sides <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=NO ALIGN=MIDDLE  ALT=\"NO\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=VA ALIGN=MIDDLE  ALT=\"VA\"   DISABLED_style= \"max-width:90%; height:auto;\" > are parallel.<BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BOV%5D=m%5BAN%5D=-5%2F3 ALIGN=MIDDLE  ALT=\"m%5BOV%5D=m%5BAN%5D=-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , sides <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OV ALIGN=MIDDLE  ALT=\"OV\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AN ALIGN=MIDDLE  ALT=\"AN\"   DISABLED_style= \"max-width:90%; height:auto;\" > are parallel.<BR>\n" );
document.write( "So, quadrilateral NOVA is a parallelogram.<BR>\n" );
document.write( "Since  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BNO%5D%2Am%5BOV%5D=%283%2F5%29%2A%28-5%2F3%29=-1 ALIGN=MIDDLE  ALT=\"m%5BNO%5D%2Am%5BOV%5D=%283%2F5%29%2A%28-5%2F3%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , sides <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=NO ALIGN=MIDDLE  ALT=\"NO\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OV ALIGN=MIDDLE  ALT=\"OV\"   DISABLED_style= \"max-width:90%; height:auto;\" > are perpendicular,<BR>\n" );
document.write( "and a parallelogram with one right angle has four right angles,<BR>\n" );
document.write( "so  quadrilateral NOVA is at least a rectangle.<BR>\n" );
document.write( "More specifically, it could also be a square,<BR>\n" );
document.write( "which is a special kind of rectangle,<BR>\n" );
document.write( "and (according to many definitions) a special kind of rhombus.<BR>\n" );
document.write( "to find out if it is a square (and a rhombus),<BR>\n" );
document.write( "we can check to see if all sides have the same length,<BR>\n" );
document.write( "or we can calculate the slopes of diagonals to see if they are perpendicular. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );