document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197616>Question 1197616</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A triangle, PQR, has P(8, 0), Q(0, −8) and point R is on the line y = x − 2. Find the area of the<BR>\n" );
document.write( "triangle PQR. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830966 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=830966\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red>24 square units</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "===========================================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Explanation:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "At first glance, this seems like a problem that might suggest an answer of \"infinitely many positive areas are possible\" due to the fact point R slides anywhere along the line y = x-2. If not infinitely many, then perhaps it suggests at least more than one possible answer. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "However, there's only one possible answer since the line y = x-2 runs parallel to line PQ.<BR>\n" );
document.write( "You can graph the items mentioned to confirm this claim, or you can use algebraic tools mentioned below.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Apply the slope formula to points P and Q to find that...<BR>\n" );
document.write( "P = (x1,y1) = (8,0)<BR>\n" );
document.write( "Q = (x2,y2) = (0,-8)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28y%5B2%5D+-+y%5B1%5D%29%2F%28x%5B2%5D+-+x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"m+=+%28y%5B2%5D+-+y%5B1%5D%29%2F%28x%5B2%5D+-+x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28-8+-+0%29%2F%280+-+8%29 ALIGN=MIDDLE  ALT=\"m+=+%28-8+-+0%29%2F%280+-+8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28-8%29%2F%28-8%29 ALIGN=MIDDLE  ALT=\"m+=+%28-8%29%2F%28-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+1 ALIGN=MIDDLE  ALT=\"m+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The slope of line PQ is 1, which matches with the slope of y = x-2, aka y = 1x-2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Either line PQ is parallel to line y = x-2, or we're talking about the same line.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "But there's no way they could be the same line because the y intercept of line PQ is Q(0,-8). <BR>\n" );
document.write( "In contrast, the y intercept of y = x-2 is (0,-2)<BR>\n" );
document.write( "Therefore, these two lines are parallel and never intersect.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Side note: parallel lines have equal slopes, but different y intercepts.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "What we can do is rotate the page so that line PQ is horizontal. <BR>\n" );
document.write( "The height of the triangle runs perpendicular from line PQ to line y = x-2. <BR>\n" );
document.write( "This height does not change as the distance between the parallel lines remains constant. Think of railroad tracks being the same width apart all throughout.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So far we've established that there's only one possible height for this triangle when we treat PQ as the base.<BR>\n" );
document.write( "This leads to exactly one area value (even if R is sliding around wherever you want).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's use the distance formula to find the length of side PQ<BR>\n" );
document.write( "(x1,y1) = (8,0) and (x2,y2) = (0,-8)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+%288-0%29%5E2+%2B+%280-%28-8%29%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%288-0%29%5E2+%2B+%280-%28-8%29%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+%288-0%29%5E2+%2B+%280%2B8%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%288-0%29%5E2+%2B+%280%2B8%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+%288%29%5E2+%2B+%288%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%288%29%5E2+%2B+%288%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+64+%2B+64+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+64+%2B+64+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+128+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+128+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%2864%2A2%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%2864%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%2864%29%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%2864%29%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+8%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"d+=+8%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Side PQ is exactly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"8%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > units long, which I'll treat as the base of triangle PQR. If needed, rotate the page so PQ is horizontal.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Next, we'll need to find the height perpendicular to side PQ.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's find the equation of the line perpendicular to y = x-2 that goes through P(8,0)<BR>\n" );
document.write( "I'll apply point-slope form.<BR>\n" );
document.write( "y - y1 = m(x - x1)<BR>\n" );
document.write( "y - 0 = -1(x - 8)<BR>\n" );
document.write( "y = -x + 8\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This perpendicular line (y = -x+8) intersects line y = x-2 at some point, why not call it R.<BR>\n" );
document.write( "Use whichever method you prefer to find R(5,3)<BR>\n" );
document.write( "This will make triangle PQR a right triangle with the 90 degree angle at point P.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then use the distance formula to find the length of side PR<BR>\n" );
document.write( "I'll skip steps, but you should get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"3%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can now compute the area of triangle PQR.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area+=+0.5%2Abase%2Aheight ALIGN=MIDDLE  ALT=\"area+=+0.5%2Abase%2Aheight\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area+=+0.5%2APQ%2APR ALIGN=MIDDLE  ALT=\"area+=+0.5%2APQ%2APR\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area+=+0.5%2A8%2Asqrt%282%29%2A3%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"area+=+0.5%2A8%2Asqrt%282%29%2A3%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area+=+12%2Asqrt%282%2A2%29 ALIGN=MIDDLE  ALT=\"area+=+12%2Asqrt%282%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area+=+12%2Asqrt%284%29 ALIGN=MIDDLE  ALT=\"area+=+12%2Asqrt%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area+=+12%2A2 ALIGN=MIDDLE  ALT=\"area+=+12%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area+=+24 ALIGN=MIDDLE  ALT=\"area+=+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The area of triangle PQR is <font color=red>24 square units</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This applies for any triangle where R can slide around freely along y = x-2, and not just a right triangle. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You can use software tools like GeoGebra to confirm the answer. I use it all the time and recommend it to any geometry or algebra students.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );