document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197253>Question 1197253</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the distance of the point (12,7)from the line through (5,3), (2,-3) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830447 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=830447\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Let's find the slope of the line through (5,3) and (2,-3)<BR>\n" );
document.write( "(x1,y1) = (5,3)<BR>\n" );
document.write( "(x2,y2) = (2,-3)\r<BR>\n" );
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document.write( "m = slope\r<BR>\n" );
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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" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28-3+-+3%29%2F%282+-+5%29 ALIGN=MIDDLE  ALT=\"m+=+%28-3+-+3%29%2F%282+-+5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28-6%29%2F%28-3%29 ALIGN=MIDDLE  ALT=\"m+=+%28-6%29%2F%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+2 ALIGN=MIDDLE  ALT=\"m+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now apply point-slope form<BR>\n" );
document.write( "y-y1 = m(x - x1)<BR>\n" );
document.write( "y-3 = 2(x - 5)<BR>\n" );
document.write( "y-3 = 2x - 10<BR>\n" );
document.write( "y = 2x - 10+3<BR>\n" );
document.write( "y = 2x - 7<BR>\n" );
document.write( "This is in y = mx+b form\r<BR>\n" );
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document.write( "Let's get everything to one side<BR>\n" );
document.write( "y = 2x - 7<BR>\n" );
document.write( "0 = 2x - 7 - y<BR>\n" );
document.write( "0 = 2x - y - 7<BR>\n" );
document.write( "2x-y-7 = 0<BR>\n" );
document.write( "This is now in the form Ax+By+C=0 with A = 2, B = -1, C = -7\r<BR>\n" );
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document.write( "Now we'll use the formula discussed here<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line\">https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line</a><BR>\n" );
document.write( "I'll use slightly different notation<BR>\n" );
document.write( "The A,B,C will be the same<BR>\n" );
document.write( "Instead of (x0,y0) I'll refer to the point not on the line as (p,q)\r<BR>\n" );
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document.write( "The distance from (p,q) to the line Ax+By+C = 0 is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+%28abs%28Ap%2BBq%2BC%29%29%2F%28sqrt%28A%5E2%2BB%5E2%29%29 ALIGN=MIDDLE  ALT=\"d+=+%28abs%28Ap%2BBq%2BC%29%29%2F%28sqrt%28A%5E2%2BB%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where point (p,q) is not on the line mentioned.\r<BR>\n" );
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document.write( "We'll plug in A = 2, B = -1, C = -7<BR>\n" );
document.write( "And also p = 12 and q = 7 from the point (12,7)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+%28abs%28Ap%2BBq%2BC%29%29%2F%28sqrt%28A%5E2%2BB%5E2%29%29 ALIGN=MIDDLE  ALT=\"d+=+%28abs%28Ap%2BBq%2BC%29%29%2F%28sqrt%28A%5E2%2BB%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+%28abs%282%2A12%2B%28-1%29%2A7%2B%28-7%29%29%29%2F%28sqrt%282%5E2%2B%28-1%29%5E2%29%29 ALIGN=MIDDLE  ALT=\"d+=+%28abs%282%2A12%2B%28-1%29%2A7%2B%28-7%29%29%29%2F%28sqrt%282%5E2%2B%28-1%29%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+%28abs%2824-7-7%29%29%2F%28sqrt%285%29%29 ALIGN=MIDDLE  ALT=\"d+=+%28abs%2824-7-7%29%29%2F%28sqrt%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+%28abs%2810%29%29%2F%28sqrt%285%29%29 ALIGN=MIDDLE  ALT=\"d+=+%28abs%2810%29%29%2F%28sqrt%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+10%2F%28sqrt%285%29%29 ALIGN=MIDDLE  ALT=\"d+=+10%2F%28sqrt%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+%2810%2Asqrt%285%29%29%2F%28sqrt%285%29%2Asqrt%285%29%29 ALIGN=MIDDLE  ALT=\"d+=+%2810%2Asqrt%285%29%29%2F%28sqrt%285%29%2Asqrt%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply top and bottom by sqrt(5) to rationalize the denominator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+%2810%2Asqrt%285%29%29%2F%28%28sqrt%285%29%29%5E2%29 ALIGN=MIDDLE  ALT=\"d+=+%2810%2Asqrt%285%29%29%2F%28%28sqrt%285%29%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+%2810%2Asqrt%285%29%29%2F5 ALIGN=MIDDLE  ALT=\"d+=+%2810%2Asqrt%285%29%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+2%2Asqrt%285%29 ALIGN=MIDDLE  ALT=\"d+=+2%2Asqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the exact distance from (12,7) to the line 2x-y-7 = 0 which goes through (5,3) and (2,-3)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2\sqrt{5} \approx 4.472136\">\r<BR>\n" );
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document.write( "-------------------------------------------\r<BR>\n" );
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document.write( "Verification:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/i9IXZkA.png\"><BR>\n" );
document.write( "I used GeoGebra to make this image. \r<BR>\n" );
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document.write( "The red line 2x-y-7 = 0 was what we found earlier through (5,3) and (2,-3)<BR>\n" );
document.write( "The black line x+2y-26 = 0 is perpendicular to the red line, and it goes through point A(12,7)<BR>\n" );
document.write( "Use whichever method you prefer to solve that system of 2 equations to find the red and black line intersect at D(8,9)<BR>\n" );
document.write( "Then use the distance formula to find the distance from A(12,7) to D(8,9) is exactly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%285%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > units.\r<BR>\n" );
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document.write( "Side note: <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%285%29+=+sqrt%282%5E2%29%2Asqrt%285%29+=+sqrt%284%29%2Asqrt%285%29+=+sqrt%284%2A5%29+=+sqrt%2820%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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