document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197813>Question 1197813</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The line joining ( -8, 1 ) and ( 11, 7 ) is divided into four equal parts. Find the points of division. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #831242 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=831242\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Refer to this lesson<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.algebra.com/algebra/homework/formulas/split-segment-n-pieces.lesson\">https://www.algebra.com/algebra/homework/formulas/split-segment-n-pieces.lesson</a><BR>\n" );
document.write( "because I'll be using the formula mentioned there. I show how the formula is derived.\r<BR>\n" );
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document.write( "In this case,<BR>\n" );
document.write( "(a,b) = (-8,1)<BR>\n" );
document.write( "(c,d) = (11,7)<BR>\n" );
document.write( "are the two endpoints\r<BR>\n" );
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document.write( "We want to split the segment into n = 4 pieces.\r<BR>\n" );
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document.write( "So<BR>\n" );
document.write( "deltaX = (1/n)*|c-a|<BR>\n" );
document.write( "deltaX = (1/4)*|11-(-8)|<BR>\n" );
document.write( "deltaX = 19/4<BR>\n" );
document.write( "deltaX = 4.75<BR>\n" );
document.write( "This is the increment we move along the horizontal axis when determining each cutoff point.<BR>\n" );
document.write( "Furthermore,<BR>\n" );
document.write( "xm = a + m*deltaX<BR>\n" );
document.write( "xm = -8 + m*4.75<BR>\n" );
document.write( "xm = 4.75m - 8<BR>\n" );
document.write( "where m is an integer in the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C+m+%3C+4 ALIGN=MIDDLE  ALT=\"0+%3C+m+%3C+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "When I write xm, I really mean \"x subscript m\" or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Bm%5D ALIGN=MIDDLE  ALT=\"x%5Bm%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "And<BR>\n" );
document.write( "deltaY = (1/n)*|d-b|<BR>\n" );
document.write( "deltaY = (1/4)*|7-1|<BR>\n" );
document.write( "deltaY = 6/4<BR>\n" );
document.write( "deltaY = 1.5<BR>\n" );
document.write( "This is the increment we move along the vertical axis when determining each cutoff point.<BR>\n" );
document.write( "Furthermore,<BR>\n" );
document.write( "ym = b + m*deltaY<BR>\n" );
document.write( "ym = 1 + m*1.5<BR>\n" );
document.write( "ym = 1.5m + 1\r<BR>\n" );
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document.write( "------------------------------------------------------------\r<BR>\n" );
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document.write( "We have this so far<BR>\n" );
document.write( "xm = 4.75m - 8<BR>\n" );
document.write( "ym = 1.5m + 1\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "(xm,ym) = (4.75m-8, 1.5m+1)<BR>\n" );
document.write( "where m is an integer such that 0 < m < 4<BR>\n" );
document.write( "This represents the general form of our cutoff points.\r<BR>\n" );
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document.write( "We'll plug in integer values from m = 1 to m = 3.\r<BR>\n" );
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document.write( "For example, if m = 1, then...<BR>\n" );
document.write( "xm = 4.75m-8<BR>\n" );
document.write( "x1 = 4.75*1-8<BR>\n" );
document.write( "x1 = 4.75-8<BR>\n" );
document.write( "x1 = -3.25<BR>\n" );
document.write( "ym = 1.5m+1<BR>\n" );
document.write( "y1 = 1.5*1+1<BR>\n" );
document.write( "y1 = 1.5+1<BR>\n" );
document.write( "y1 = 2.5<BR>\n" );
document.write( "This gives one cutoff point at (x1,y1) = <font color=red>(-3.25,2.5)</font><BR>\n" );
document.write( "This is <font color=red>point B</font> shown below.\r<BR>\n" );
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document.write( "If m = 2, then,<BR>\n" );
document.write( "xm = 4.75m-8<BR>\n" );
document.write( "x2 = 4.75*2-8<BR>\n" );
document.write( "x2 =9.5-8<BR>\n" );
document.write( "x2 = 1.5<BR>\n" );
document.write( "ym = 1.5m+1<BR>\n" );
document.write( "y2 = 1.5*2+1<BR>\n" );
document.write( "y2 = 3+1<BR>\n" );
document.write( "y2 = 4<BR>\n" );
document.write( "This gives another cutoff point at (x2,y2) = <font color=red>(1.5,4)</font><BR>\n" );
document.write( "This is <font color=red>point C</font> shown below.\r<BR>\n" );
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document.write( "Here's a table of values to keep things organized. <BR>\n" );
document.write( "Spreadsheet software is recommended to not only keep things organized, but to also quickly compute each (xm,ym) cutoff point.<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >m</TD><TD >xm = 4.75m-8</TD><TD >ym = 1.5m+1</TD><TD >(xm,ym)</TD><TD >Point</TD></tr><tr><TD >1</TD><TD >-3.25</TD><TD >2.5</TD><TD ><font color=red>(-3.25,2.5)</font></TD><TD ><font color=red>B</font></TD></tr><tr><TD >2</TD><TD >1.5</TD><TD >4</TD><TD ><font color=red>(1.5,4)</font></TD><TD ><font color=red>C</font></TD></tr><tr><TD >3</TD><TD >6.25</TD><TD >5.5</TD><TD ><font color=red>(6.25,5.5)</font></TD><TD ><font color=red>D</font></TD></tr>\n" );
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document.write( "Here's what the graph looks like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C15%2C-3%2C8%2C%0D%0Agraph%28500%2C500%2C-10%2C15%2C-3%2C8%2C0%29%2C%0D%0Aline%28-8%2C1%2C11%2C7%29%2C%0D%0Acircle%28-8%2C1%2C0%29%2C%0D%0Acircle%28-8%2C1%2C0.1%29%2C%0D%0Acircle%28-8%2C1%2C0.2%29%2C%0D%0Ared%28circle%28-3.25%2C2.5%2C0%29%29%2C%0D%0Ared%28circle%28-3.25%2C2.5%2C0.1%29%29%2C%0D%0Ared%28circle%28-3.25%2C2.5%2C0.2%29%29%2C%0D%0Ared%28circle%281.5%2C4%2C0%29%29%2C%0D%0Ared%28circle%281.5%2C4%2C0.1%29%29%2C%0D%0Ared%28circle%281.5%2C4%2C0.2%29%29%2C%0D%0Ared%28circle%286.25%2C5.5%2C0%29%29%2C%0D%0Ared%28circle%286.25%2C5.5%2C0.1%29%29%2C%0D%0Ared%28circle%286.25%2C5.5%2C0.2%29%29%2C%0D%0Acircle%2811%2C7%2C0%29%2C%0D%0Acircle%2811%2C7%2C0.1%29%2C%0D%0Acircle%2811%2C7%2C0.2%29%2C%0D%0Alocate%28-8%2C1-0.25%2C%22A%22%29%2C%0D%0Ared%28locate%28-3.25%2C2.5-0.25%2C%22B%22%29%29%2C%0D%0Ared%28locate%281.5%2C4-0.25%2C%22C%22%29%29%2C%0D%0Ared%28locate%286.25%2C5.5-0.25%2C%22D%22%29%29%2C%0D%0Alocate%2811%2C7-0.25%2C%22E%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The answer points are marked in red.\r<BR>\n" );
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document.write( "A = (-8,1) is one endpoint<BR>\n" );
document.write( "<font color=red>B = (-3.25,2.5)<BR>\n" );
document.write( "C = (1.5,4)<BR>\n" );
document.write( "D = (6.25,5.5)</font><BR>\n" );
document.write( "E = (11,7) is the other endpoint\r<BR>\n" );
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document.write( "------------------------------------------------------------\r<BR>\n" );
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document.write( "To confirm the answers, use the distance formula to show the following<BR>\n" );
document.write( "AE = sqrt(397)<BR>\n" );
document.write( "AB = sqrt(24.8125)<BR>\n" );
document.write( "BC = sqrt(24.8125)<BR>\n" );
document.write( "CD = sqrt(24.8125)<BR>\n" );
document.write( "DE = sqrt(24.8125)<BR>\n" );
document.write( "This will show that AB=BC=CD=DE and that segment AE has been cut into four equal pieces. <BR>\n" );
document.write( "I'll let the student do these steps for the confirmation.\r<BR>\n" );
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document.write( "Note:<BR>\n" );
document.write( "AE = 4*AB<BR>\n" );
document.write( "sqrt(397) = 4*sqrt(24.8125)\r<BR>\n" );
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document.write( "Another way to confirm the answers is to use something like GeoGebra. \r<BR>\n" );
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document.write( "Yet another confirmation method is to use the midpoint formula on endpoints A and E, and it will pinpoint where C is located. <BR>\n" );
document.write( "Then use the midpoint formula on A and C to find where B is located.<BR>\n" );
document.write( "Finally, use the midpoint formula on C and E to find where D is located.<BR>\n" );
document.write( "This repeated midpoint trick only works when there are 2^k equal pieces, where k is some positive integer.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );