document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1111745>Question 1111745</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the perpendicular distance between the line y=(5/13)x + 5 and the origin. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #726758 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=726758\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Distance from a point to a line is always measured along a perpendicular, of course.<BR>\n" );
document.write( "If you were at point P in the sketch below,along what path would you measure your distance to the pool?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-6%2C6%2C-1%2C11%2C%0D%0Ablue%28line%28-5%2C1%2C0%2C4%29%29%2C%0D%0Agreen%28line%28-5%2C1%2C0%2C1%29%29%2C%0D%0Agreen%28rectangle%280%2C1%2C-0.5%2C1.5%29%29%2C%0D%0Ared%28line%28-5%2C1%2C0%2C8%29%29%2C%0D%0Ablue%28rectangle%280%2C0%2C5%2C10%29%29%2C%0D%0Alocate%282%2C5.5%2Cblue%28pool%29%29%2C%0D%0Alocate%28-5.1%2C0.9%2CP%29%2C%0D%0Alocate%28-2.5%2C1%2Cd%29%2C%0D%0Alocate%28-2.5%2C2.5%2Ce%29%2C%0D%0Alocate%28-2.5%2C4.5%2Cf%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > along what path would you measure your distance to the pool, d, e, or f?<BR>\n" );
document.write( " <BR>\n" );
document.write( "MY GUESS AS TO THE EXPECTED SOLUTION:<BR>\n" );
document.write( "Your teacher (or textbook) has probably given you the formula<BR>\n" );
document.write( "to calculate the distance between a point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%5BP%5D%2Cy%5BP%5D%29 ALIGN=MIDDLE  ALT=\"P%28x%5BP%5D%2Cy%5BP%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and a line with equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%2Bby%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%2Bby%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=distance ALIGN=MIDDLE  ALT=\"distance\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28ax%5BP%5D%2Bby%5BP%5D%2Bc%29%2Fsqrt%28a%5E2%2Bb%5E2%29 ALIGN=MIDDLE  ALT=\"abs%28ax%5BP%5D%2Bby%5BP%5D%2Bc%29%2Fsqrt%28a%5E2%2Bb%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "In this case, your point is the origin <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=O%280%2C0%29 ALIGN=MIDDLE  ALT=\"O%280%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5BP%5D=y%5BP%5D=0 ALIGN=MIDDLE  ALT=\"x%5BP%5D=y%5BP%5D=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%285%2F13%29x+%2B+5 ALIGN=MIDDLE  ALT=\"y=%285%2F13%29x+%2B+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be transformed,<BR>\n" );
document.write( "by rearranging and multiplying both sides of the equal sign time <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13 ALIGN=MIDDLE  ALT=\"13\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "to get the equivalent equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x-13y%2B65=0 ALIGN=MIDDLE  ALT=\"5x-13y%2B65=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=5 ALIGN=MIDDLE  ALT=\"a=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-13 ALIGN=MIDDLE  ALT=\"b=-13\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=65 ALIGN=MIDDLE  ALT=\"c=65\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "So, in this case,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=distance ALIGN=MIDDLE  ALT=\"distance\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=65%2Fsqrt%285%5E2%2B%28-13%29%5E2%29 ALIGN=MIDDLE  ALT=\"65%2Fsqrt%285%5E2%2B%28-13%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >}<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=65%2Fsqrt%2825%2B169%29 ALIGN=MIDDLE  ALT=\"65%2Fsqrt%2825%2B169%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >}<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=65%2Fsqrt%28194%29 ALIGN=MIDDLE  ALT=\"65%2Fsqrt%28194%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The elegant way to express the exact value is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=distance=65sqrt%28194%29%2F194 ALIGN=MIDDLE  ALT=\"distance=65sqrt%28194%29%2F194\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The approximate value for that irrational number<BR>\n" );
document.write( "is the never ending, non-repeating, decima <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22l4.666728....%22 ALIGN=MIDDLE  ALT=\"%22l4.666728....%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "ANOTHER APPROACH:<BR>\n" );
document.write( "It is easy to see that the x- and y-intercepts are the points<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28-13%2C0%29 ALIGN=MIDDLE  ALT=\"A%28-13%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%280%2C5%29 ALIGN=MIDDLE  ALT=\"B%280%2C5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If you look at the line, the axes,<BR>\n" );
document.write( "and the path along which you would measure the distance to the origin,<BR>\n" );
document.write( "you see right triangles:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28600%2C300%2C-16%2C4%2C-3%2C8%2Cgrid%280%29%2C%0D%0Agreen%28triangle%280%2C0%2C-1.675%2C4.356%2C0%2C5%29%29%2C%0D%0Aline%28-16%2C-1.154%2C5%2C6.923%29%2Clocate%280.1%2C0%2CO%29%0D%0Alocate%28-13.2%2C0.6%2CA%29%2Clocate%280.1%2C5%2CB%29%2C%0D%0Alocate%28-1.85%2C4.9%2CC%29%2Clocate%28-1.3%2C3.9%2Cgreen%28d%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "When people see right triangles they think about<BR>\n" );
document.write( "trigonometric ratios,<BR>\n" );
document.write( "or the Pythagorean theorem,<BR>\n" );
document.write( "or similar triangles.<BR>\n" );
document.write( "Each of those ideas can also lead to a solution,<BR>\n" );
document.write( "as another tutor showed you for trigonometric ratios.<BR>\n" );
document.write( " <BR>\n" );
document.write( "USING THE PYTHAGOREAN THEOREM:<BR>\n" );
document.write( "The most obvious right triangle in the sketch above is big triangle ABO ,<BR>\n" );
document.write( "with hypotenuse <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=sqrt%2813%5E2%2B5%5E2%29=sqrt%28169%2B25%29=sqrt%28194%29 ALIGN=MIDDLE  ALT=\"AB=sqrt%2813%5E2%2B5%5E2%29=sqrt%28169%2B25%29=sqrt%28194%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "calculated using the Pythagorean theorem.<BR>\n" );
document.write( "The area of ABO is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=5%2A13%2F2 ALIGN=MIDDLE  ALT=\"area=5%2A13%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "calculated as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=base%2Aheight%2F2 ALIGN=MIDDLE  ALT=\"area=base%2Aheight%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "using AO as the base and OB as the height.<BR>\n" );
document.write( "As we know that the distance <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28d%29 ALIGN=MIDDLE  ALT=\"green%28d%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "is measured along line OC, perpendicular to AB,<BR>\n" );
document.write( "we could also calculate the area as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=AB%2Ad%2F2 ALIGN=MIDDLE  ALT=\"area=AB%2Ad%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB ALIGN=MIDDLE  ALT=\"AB\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the base and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28d%29=OC ALIGN=MIDDLE  ALT=\"green%28d%29=OC\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the height.<BR>\n" );
document.write( "Substituting, and combining both ways of calculating area<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28194%29d%2F2=5%2A13%2F2 ALIGN=MIDDLE  ALT=\"sqrt%28194%29d%2F2=5%2A13%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28194%29d=5%2A13 ALIGN=MIDDLE  ALT=\"sqrt%28194%29d=5%2A13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=5%2A13%2Fsqrt%28194%29=65%2Fsqrt%28194%29=65sqrt%28194%29%2F194=aproximately14.67 ALIGN=MIDDLE  ALT=\"d=5%2A13%2Fsqrt%28194%29=65%2Fsqrt%28194%29=65sqrt%28194%29%2F194=aproximately14.67\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "USING SIMILAR TRIANGLES:<BR>\n" );
document.write( "Triangles ABO and BCO are both right triangles,<BR>\n" );
document.write( "and they both have the same angle at B,<BR>\n" );
document.write( "so they are similar triangles,<BR>\n" );
document.write( "with BCO being a scaled-down version of ABO.<BR>\n" );
document.write( "The ratio of corresponding sides is the scale factor,<BR>\n" );
document.write( "the same for the long leg as for the hypotenuse<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%2F13=5%2Fsqrt%28194%29 ALIGN=MIDDLE  ALT=\"d%2F13=5%2Fsqrt%28194%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=5%2A13%2Fsqrt%28194%29=65%2Fsqrt%28194%29=65sqrt%28194%29%2F194=aproximately14.67 ALIGN=MIDDLE  ALT=\"d=5%2A13%2Fsqrt%28194%29=65%2Fsqrt%28194%29=65sqrt%28194%29%2F194=aproximately14.67\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );