document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1012777>Question 1012777</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show algebraically that this triangle is a right triangle. The triangle has the vertices P(-4,-3) Q(2,5) R(4,1) <BR>\n" );
document.write( "Find the midpoint of the hypotenuse.<BR>\n" );
document.write( "Show that this midpoint is equidistant from each of the vertices<BR>\n" );
document.write( "Thanks! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #628875 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=628875\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are two ways to show that the triangle is a right triangle.<BR>\n" );
document.write( "You could find that two of the sides are perpendicular, or<BR>\n" );
document.write( "you could find that the square of the length of one side is the sum of the squares of the lengths of the other two sides.<BR>\n" );
document.write( " <BR>\n" );
document.write( "USING THE LENGTHS OF THE SIDES:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ%5E2=%282-%28-4%29%29%5E2%2B%285-%28-3%29%29%5E2=%282%2B4%29%5E2%2B%285%2B3%29%5E2=6%5E2%2B8%5E2=36%2B64=100 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=QR%5E2=%284-2%29%5E2%2B%285-1%29%5E2=2%5E2%2B4%5E2=4%2B16=20 ALIGN=MIDDLE  ALT=\"QR%5E2=%284-2%29%5E2%2B%285-1%29%5E2=2%5E2%2B4%5E2=4%2B16=20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PR%5E2=%284-%28-4%29%29%5E2%2B%281-%28-3%29%29%5E2=%284%2B4%29%5E2%2B%281%2B3%29%5E2=8%5E2%2B4%5E2=64%2B16=80 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=QR%5E2%2BPR%5E2=20%2B80=100=PQ%5E2 ALIGN=MIDDLE  ALT=\"QR%5E2%2BPR%5E2=20%2B80=100=PQ%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, by the converse of the Pythagorean theorem,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=QR ALIGN=MIDDLE  ALT=\"QR\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PR ALIGN=MIDDLE  ALT=\"PR\"   DISABLED_style= \"max-width:90%; height:auto;\" > are the legs of a right triangle with hypotenuse <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ ALIGN=MIDDLE  ALT=\"PQ\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "PROVING THAT TWO SIDES ARE PERPENDICULAR:<BR>\n" );
document.write( "IF you have learned about slope of a line,<BR>\n" );
document.write( "you may also have learned that if two lines have slopes whose product is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , those lines are perpendicular.<BR>\n" );
document.write( "Slope of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PR ALIGN=MIDDLE  ALT=\"PR\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281-%28-3%29%29%2F%284-%28-4%29%29=%281%2B3%29%2F%284%2B4%29=4%2F8=1%2F2 ALIGN=MIDDLE  ALT=\"%281-%28-3%29%29%2F%284-%28-4%29%29=%281%2B3%29%2F%284%2B4%29=4%2F8=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Slope of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=QR ALIGN=MIDDLE  ALT=\"QR\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285-1%29%2F%282-4%29=4%2F%28-2%29=-2 ALIGN=MIDDLE  ALT=\"%285-1%29%2F%282-4%29=4%2F%28-2%29=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The product of the slopes is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%28-2%29=-1 ALIGN=MIDDLE  ALT=\"%281%2F2%29%28-2%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PR ALIGN=MIDDLE  ALT=\"PR\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=QR ALIGN=MIDDLE  ALT=\"QR\"   DISABLED_style= \"max-width:90%; height:auto;\" > are perpendicular,<BR>\n" );
document.write( "which means that triangle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQR ALIGN=MIDDLE  ALT=\"PQR\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a right triangle,<BR>\n" );
document.write( "with a right angle at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > and hypotenuse <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ ALIGN=MIDDLE  ALT=\"PQ\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "MIDPOINT OF THE HYPOTENUSE:<BR>\n" );
document.write( "The coordinates of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%28x%5BM%5D%2Cy%5BM%5D%29 ALIGN=MIDDLE  ALT=\"M%28x%5BM%5D%2Cy%5BM%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the midpoint of hypotenuse <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ ALIGN=MIDDLE  ALT=\"PQ\"   DISABLED_style= \"max-width:90%; height:auto;\" > , are found by averaging the coordinates of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Q ALIGN=MIDDLE  ALT=\"Q\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5BM%5D=%28x%5BP%5D%2Bx%5BQ%5D%29%2F2=%28-4%2B2%29%2F2_-2%2F2=-1 ALIGN=MIDDLE  ALT=\"x%5BM%5D=%28x%5BP%5D%2Bx%5BQ%5D%29%2F2=%28-4%2B2%29%2F2_-2%2F2=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5BM%5D=%28y%5BP%5D%2By%5BQ%5D%29%2F2=%28-3%2B5%29%2F2_2%2F2=1 ALIGN=MIDDLE  ALT=\"y%5BM%5D=%28y%5BP%5D%2By%5BQ%5D%29%2F2=%28-3%2B5%29%2F2_2%2F2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "DISTANCES FROM <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%28-1%2C1%29 ALIGN=MIDDLE  ALT=\"M%28-1%2C1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > TO <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Q ALIGN=MIDDLE  ALT=\"Q\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "To show that the distances are the same, we can just show that their squares are the same.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PM%5E2=%28-4-%28-1%29%29%5E2%2B%28-3-1%29%5E2=%28-4%2B1%29%5E2%2B%28-4%29%5E2=%28-3%29%5E2%2B16=9%2B16=25 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=QM%5E2=%282-%28-1%29%29%5E2%2B%285-1%29%5E2=%282%2B1%29%5E2%2B4%5E2=3%5E2%2B16=9%2B16=25 ALIGN=MIDDLE  ALT=\"QM%5E2=%282-%28-1%29%29%5E2%2B%285-1%29%5E2=%282%2B1%29%5E2%2B4%5E2=3%5E2%2B16=9%2B16=25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=RM%5E2=%284-%28-1%29%29%5E2%2B%281-1%29%5E2=%284%2B1%29%5E2%2B0%5E2=5%5E2%2B0=25%2B0=25 ALIGN=MIDDLE  ALT=\"RM%5E2=%284-%28-1%29%29%5E2%2B%281-1%29%5E2=%284%2B1%29%5E2%2B0%5E2=5%5E2%2B0=25%2B0=25\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, the distances from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M ALIGN=MIDDLE  ALT=\"M\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Q ALIGN=MIDDLE  ALT=\"Q\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > are all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2825%29=5 ALIGN=MIDDLE  ALT=\"sqrt%2825%29=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );