document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=547990>Question 547990</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the shortest distance between the point (4,2) and the line <BR>\n" );
document.write( "y = 3/4x - 31. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #356676 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=356676\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=Distance_between_a_point_and_a_line.solver>Finding a distance between a point given by coordinates (x, y)  and a line given by equation y=ax+b</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>We want to find the perpendicular distance between a point given by coordinates (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "  and a line given by equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0.75%2Ax%2B-31 ALIGN=MIDDLE  ALT=\"y=0.75%2Ax%2B-31\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, let's draw a diagram of general situation with point <b>P</b> (xo, yo) and <BR>\n" );
document.write( "  line <b>L:</b> y= a.x + b. The required distance is <b>PC</b>. (in the diagram below)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+200%2C+200%2C+-2%2C+8%2C+-2%2C+8%2C+line%28-2%2C-2%2C8%2C8%29%2Cline%280%2C0%2C0%2C4%29%2Cline%280%2C4%2C4%2C4%29%2Cgreen%28line%280%2C4%2C2%2C2%29%29%2Clocate%280%2C5%2CP%28xo%2Cyo%29%29%2Clocate%280%2C0%2CA%28x1%2Cy1%29%29%2C+locate%284%2C4%2CB%28x2%2Cy2%29%29%2Clocate%282%2C2%2CC%29%2Clocate%286%2C6%2CL%29%2C++red%28circle%280%2C0%2C.1%29%29%2Cred%28circle%284%2C4%2C.1%29%29+%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <b>Methodology</b><BR>\n" );
document.write( "  We will first find the vertices of the triangle in order to get the side lengths and then by applying <BR>\n" );
document.write( "  <b><A  rel=nofollow HREF=Sine_rule.wikipedia>Sine Rule</A></b> on <A  rel=nofollow HREF=Special_right_triangles.wikipedia>right angle triangle</A> <b>PAB</b> and <b>PBC</b> we will calculate the desired distance <b>PC</b>. <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <b>Step1</b><BR>\n" );
document.write( "  Calculation of the vertices of triangle PAB:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Draw a vertical line passing through the point 'P'. This line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > will cut the given line 'L' <BR>\n" );
document.write( "  at point 'A'. The X coordinate of A(<b>x1</b>) will be same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xo=4 ALIGN=MIDDLE  ALT=\"xo=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. To find the Y-coordinate of<BR>\n" );
document.write( "  'A' we will use the fact that <b>point 'A' lies on the given line 'L' and satisfies the equation <BR>\n" );
document.write( "  of the line 'L' </b>.<BR>\n" );
document.write( "  Now, plug this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x1=4 ALIGN=MIDDLE  ALT=\"x1=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > in to the equation of line: y=0.75*x+-31 <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y1=0.75%2A4+%2B-31 ALIGN=MIDDLE  ALT=\"y1=0.75%2A4+%2B-31\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y1=-28 ALIGN=MIDDLE  ALT=\"y1=-28\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Hence, Point (A)(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x1=4 ALIGN=MIDDLE  ALT=\"x1=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y1=-28 ALIGN=MIDDLE  ALT=\"y1=-28\"   DISABLED_style= \"max-width:90%; height:auto;\" >) <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Similarly, <BR>\n" );
document.write( "  Draw a horizontal line passing through the point 'P'. This line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2 ALIGN=MIDDLE  ALT=\"y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > will cut the given line 'L' <BR>\n" );
document.write( "  at point 'B'. The Y coordinate of B(<b>y2</b>) will be same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=yo=2 ALIGN=MIDDLE  ALT=\"yo=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. To find the X-coordinate of<BR>\n" );
document.write( "  B we will use the fact that <b>point 'B' lies on the given line 'L' and satisfies the equation <BR>\n" );
document.write( "  of the line 'L' </b>.<BR>\n" );
document.write( "  Now, plug this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y2=2 ALIGN=MIDDLE  ALT=\"y2=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > in to the equation of line: y=0.75*x+-31<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2=0.75%2Ax2%2B-31 ALIGN=MIDDLE  ALT=\"2=0.75%2Ax2%2B-31\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x2=+%282--31%29%2F0.75 ALIGN=MIDDLE  ALT=\"x2=+%282--31%29%2F0.75\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x2=44 ALIGN=MIDDLE  ALT=\"x2=44\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Hence, Point (B)(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x2=44 ALIGN=MIDDLE  ALT=\"x2=44\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y2=2 ALIGN=MIDDLE  ALT=\"y2=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Now, we have all the vertices of the triangle PAB<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <b>Step2</b><BR>\n" );
document.write( "  Calculation of the side lengths using <A  rel=nofollow HREF=Distance_formula.wikipedia>distance formula</A>: <BR>\n" );
document.write( "  <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;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Hence, The side lengths PA, PB and AB are<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PA=30 ALIGN=MIDDLE  ALT=\"PA=30\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PB=40 ALIGN=MIDDLE  ALT=\"PB=40\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=50 ALIGN=MIDDLE  ALT=\"AB=50\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "   <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <b>Step3</b><BR>\n" );
document.write( "  Apply <b>Sine rule</b> on common angle <b>B</b> in triangle <b>PAB</b> and triangle <b>PBC</b>.<BR>\n" );
document.write( "  Both triangle <b>PAB</b> and triangle <b>PBC</b> are <A  rel=nofollow HREF=Special_right_triangles.wikipedia>right angle triangle</A> and points 'A', 'B' and 'C' lay on the given line L. <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Sin%28B%29=+AP%2FAB=PC%2FBP ALIGN=MIDDLE  ALT=\"Sin%28B%29=+AP%2FAB=PC%2FBP\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PC=%28AP%2ABP%29%2FAB=+24 ALIGN=MIDDLE  ALT=\"PC=%28AP%2ABP%29%2FAB=+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  PC is the required perpendicular distance of the point <b>P</b> (4, 2) from line given <BR>\n" );
document.write( "  line<b>L1:</b> y=0.75*x+-31.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For better understanding of this concept, look at the Lesson based on the above concept. <BR>\n" );
document.write( "   <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Coordinate-system/Distance-of-a-Point-From-a-Line.lesson> Lesson </A></TD></TR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );