document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=174494>Question 174494</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Quadrilateral ABCD has vertices A(-1,0), B(3,3), C(6,-1), and D(2,-4). Prove that quadrilateral ABCD is a square. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #129522 by </B><A HREF=/tutors/aboutme.mpl?userid=jojo14344><B>jojo14344(1513)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jojo14344><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=129522\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> </pre><font size=4><b><BR>\n" );
document.write( "<BR>ABCD has vertices A(-1,0), B(3,3), C(6,-1), and D(2,-4)<BR>\n" );
document.write( "Let's see 3 important properties of a Square:<BR>\n" );
document.write( "1) All sides are EQUAL.<BR>\n" );
document.write( "2) The Diagonals are equal.3<BR>\n" );
document.write( "3) Each Angle is equal to 90 degress.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "To Prove the properties, let's draw the Square first with given vertices:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-6%2C8%2C-6%2C6%2Cgraph%28400%2C400%2C-6%2C8%2C-6%2C6%29%2Cline%28-1%2C0%2C3%2C3%29%2Cred%28line%283%2C3%2C6%2C-1%29%29%2Cblue%28line%286%2C-1%2C2%2C-4%29%29%2Cgreen%28line%282%2C-4%2C-1%2C0%29%29%2Clocate%28-1.3%2C.7%2CA%29%2Clocate%283.3%2C3.3%2CB%29%2Clocate%286.3%2C-1%2CC%29%2Clocate%282.3%2C-4%2CD%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "1) All SIDES ARE EQUAL.<BR>\n" );
document.write( "To prove, we use distance Formula ----><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2=%28y%5B2%5D-y%5B1%5D%29%5E2%2B%28x%5B2%5D-x%5B1%5D%29%5E2 ALIGN=MIDDLE  ALT=\"d%5E2=%28y%5B2%5D-y%5B1%5D%29%5E2%2B%28x%5B2%5D-x%5B1%5D%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB ALIGN=MIDDLE  ALT=\"AB\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5BAB%5D%5E2=%283-0%29%5E2%2B%283-%28-1%29%5E2%29=%283%5E2%2B%283%2B1%29%5E2%29=%283%5E2%2B4%5E2%29=9%2B16 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=d%5BAB%5D=sqrt%2825%29=highlight%285=d%5BAB%5D%29 ALIGN=MIDDLE  ALT=\"d%5BAB%5D=sqrt%2825%29=highlight%285=d%5BAB%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC ALIGN=MIDDLE  ALT=\"BC\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5BBC%5D%5E2=%28-1-3%29%5E2%2B%286-3%29%5E2=%28-4%5E2%2B3%5E2%29=16%2B9 ALIGN=MIDDLE  ALT=\"d%5BBC%5D%5E2=%28-1-3%29%5E2%2B%286-3%29%5E2=%28-4%5E2%2B3%5E2%29=16%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5BBC%5D=sqrt%2825%29=highlight%285=d%5BBC%5D%29 ALIGN=MIDDLE  ALT=\"d%5BBC%5D=sqrt%2825%29=highlight%285=d%5BBC%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CD ALIGN=MIDDLE  ALT=\"CD\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5BCD%5D%5E2=%28-4-%28-1%29%29%5E2%2B%282-6%29%5E2=%28-4%2B1%29%5E2%2B%28-4%29%5E2=-3%5E2%2B%28-4%29%5E2=9%2B16 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=d%5BCD%5D=sqrt%2825%29=highlight%285=d%5BCD%5D%29 ALIGN=MIDDLE  ALT=\"d%5BCD%5D=sqrt%2825%29=highlight%285=d%5BCD%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AD ALIGN=MIDDLE  ALT=\"AD\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5BAD%5D%5E2=%28-4-0%29%5E2%2B%282-%28-1%29%29%5E2=-4%5E2%2B%28-3%29%5E2=16%2B9 ALIGN=MIDDLE  ALT=\"d%5BAD%5D%5E2=%28-4-0%29%5E2%2B%282-%28-1%29%29%5E2=-4%5E2%2B%28-3%29%5E2=16%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5BAD%5D=sqrt%2825%29=highlight%285=d%5BAD%5D%29 ALIGN=MIDDLE  ALT=\"d%5BAD%5D=sqrt%2825%29=highlight%285=d%5BAD%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Therefore, you can see all SIDES are EQUAL IN LENGTH,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5BAB%5D=d%5BBC%5D=d%5BCD%5D=d%5BAD%5D=5 ALIGN=MIDDLE  ALT=\"d%5BAB%5D=d%5BBC%5D=d%5BCD%5D=d%5BAD%5D=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "It follows, as you see in the graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-6%2C8%2C-6%2C6%2Cgraph%28400%2C400%2C-6%2C8%2C-6%2C6%29%2Cline%28-1%2C0%2C3%2C3%29%2Cline%283%2C3%2C6%2C-1%29%2Cline%286%2C-1%2C2%2C-4%29%2Cline%282%2C-4%2C-1%2C0%29%2Clocate%28-1.3%2C.7%2CA%29%2Clocate%283.3%2C3.3%2CB%29%2Clocate%286.3%2C-1%2CC%29%2Clocate%282.3%2C-4%2CD%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "2) The Diagonals are EQUAL.<BR>\n" );
document.write( "The diagonals here are referred to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BD ALIGN=MIDDLE  ALT=\"BD\"   DISABLED_style= \"max-width:90%; height:auto;\" > adn <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC ALIGN=MIDDLE  ALT=\"AC\"   DISABLED_style= \"max-width:90%; height:auto;\" >. And again we use Distance formula, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2=%28y%5B2%5D-y%5B1%5D%29%5E2%2B%28x%5B2%5D-x%5B1%5D%29%5E2 ALIGN=MIDDLE  ALT=\"d%5E2=%28y%5B2%5D-y%5B1%5D%29%5E2%2B%28x%5B2%5D-x%5B1%5D%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BD ALIGN=MIDDLE  ALT=\"BD\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5BBD%5D%5E2=%28-4-3%29%5E2%2B%282-3%29%5E2=-7%5E2%2B-1%5E2=49%2B1 ALIGN=MIDDLE  ALT=\"d%5BBD%5D%5E2=%28-4-3%29%5E2%2B%282-3%29%5E2=-7%5E2%2B-1%5E2=49%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28d%5BBD%5D=sqrt%2850%29%29 ALIGN=MIDDLE  ALT=\"highlight%28d%5BBD%5D=sqrt%2850%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC ALIGN=MIDDLE  ALT=\"AC\"   DISABLED_style= \"max-width:90%; height:auto;\" >;<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5BAC%5D=%28-1-0%29%5E2%2B%286-%28-1%29%29%5E2=-1%5E2%2B%286%2B1%29%5E2=-1%5E2%2B7%5E2 ALIGN=MIDDLE  ALT=\"d%5BAC%5D=%28-1-0%29%5E2%2B%286-%28-1%29%29%5E2=-1%5E2%2B%286%2B1%29%5E2=-1%5E2%2B7%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28d%5BAC%5D=sqrt%2850%29%29 ALIGN=MIDDLE  ALT=\"highlight%28d%5BAC%5D=sqrt%2850%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Therefore: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5BBD%5D=d%5BAC%5D=sqrt%2850%29 ALIGN=MIDDLE  ALT=\"d%5BBD%5D=d%5BAC%5D=sqrt%2850%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "It satisfy the 2nd property, and as we see in the graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-6%2C8%2C-6%2C6%2Cgraph%28400%2C400%2C-6%2C8%2C-6%2C6%29%2Cline%28-1%2C0%2C3%2C3%29%2Cline%283%2C3%2C6%2C-1%29%2Cline%286%2C-1%2C2%2C-4%29%2Cline%282%2C-4%2C-1%2C0%29%2Clocate%28-1.3%2C.7%2CA%29%2Clocate%283.3%2C3.3%2CB%29%2Clocate%286.3%2C-1%2CC%29%2Clocate%282.3%2C-4%2CD%29%2Cgreen%28line%283%2C3%2C2%2C-4%29%29%2Cgreen%28line%28-1%2C0%2C6%2C-1%29%29%2Cred%28locate%282.5%2C1%2Csqrt%2850%29%29%29%2Cred%28locate%283.5%2C-.5%2Csqrt%2850%29%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "3) Each Angle is equal to 90 DEGREES<BR>\n" );
document.write( "In this property we use Equation for Right Traingles, the Pythagorean theorem.<BR>\n" );
document.write( "Just where the formula we used above is derived from.<BR>\n" );
document.write( "For Triangle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ABC ALIGN=MIDDLE  ALT=\"ABC\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC%5E2=AB%5E2%2BBC%5E2 ALIGN=MIDDLE  ALT=\"AC%5E2=AB%5E2%2BBC%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC%5E2=5%5E2%2B5%5E2 ALIGN=MIDDLE  ALT=\"AC%5E2=5%5E2%2B5%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC%5E2=25%2B25 ALIGN=MIDDLE  ALT=\"AC%5E2=25%2B25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC=sqrt%2850%29=DIAGONAL ALIGN=MIDDLE  ALT=\"AC=sqrt%2850%29=DIAGONAL\"   DISABLED_style= \"max-width:90%; height:auto;\" >, A Right Triangle!<BR>\n" );
document.write( "For Triangle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BCD ALIGN=MIDDLE  ALT=\"BCD\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BD%5E2=BC%5E2%2BCD%5E2 ALIGN=MIDDLE  ALT=\"BD%5E2=BC%5E2%2BCD%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BD%5E2=5%5E2%2B5%5E2=25%2B25 ALIGN=MIDDLE  ALT=\"BD%5E2=5%5E2%2B5%5E2=25%2B25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BD=sqrt%2850%29=DIAGONAL ALIGN=MIDDLE  ALT=\"BD=sqrt%2850%29=DIAGONAL\"   DISABLED_style= \"max-width:90%; height:auto;\" >, A Right Triangle!<BR>\n" );
document.write( "For Triangle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CDA ALIGN=MIDDLE  ALT=\"CDA\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC%5E2=AD%5E2%2BCD%5E2 ALIGN=MIDDLE  ALT=\"AC%5E2=AD%5E2%2BCD%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC%5E2=5%5E2%2B5%5E2=25%2B25 ALIGN=MIDDLE  ALT=\"AC%5E2=5%5E2%2B5%5E2=25%2B25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC=sqrt%2850%29=DIAGONAL ALIGN=MIDDLE  ALT=\"AC=sqrt%2850%29=DIAGONAL\"   DISABLED_style= \"max-width:90%; height:auto;\" >, A Right Triangle<BR>\n" );
document.write( "For Triangle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DAB ALIGN=MIDDLE  ALT=\"DAB\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BD%5E2=AD%5E2%2BAB%5E2 ALIGN=MIDDLE  ALT=\"BD%5E2=AD%5E2%2BAB%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BD%5E2=5%5E2%2B5%5E2=25%2B25 ALIGN=MIDDLE  ALT=\"BD%5E2=5%5E2%2B5%5E2=25%2B25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BD=sqrt%2850%29=DIAGONAL ALIGN=MIDDLE  ALT=\"BD=sqrt%2850%29=DIAGONAL\"   DISABLED_style= \"max-width:90%; height:auto;\" >, A Right Triangle!<BR>\n" );
document.write( "ALL satisfies the Pythagorean theorem, therefore all angles are 90 degrees:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Conclusion: ABCD that has vertices A(-1,0), B(3,3), C(6,-1), and D(2,-4) is a SQUARE.<BR>\n" );
document.write( "Thank you,<BR>\n" );
document.write( "Jojo</pre><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );