Question 1197813
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Refer to this lesson
<a 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>
because I'll be using the formula mentioned there. I show how the formula is derived.


In this case,
(a,b) = (-8,1)
(c,d) = (11,7)
are the two endpoints


We want to split the segment into n = 4 pieces.


So
deltaX = (1/n)*|c-a|
deltaX = (1/4)*|11-(-8)|
deltaX = 19/4
deltaX = 4.75
This is the increment we move along the horizontal axis when determining each cutoff point.
Furthermore,
xm = a + m*deltaX
xm = -8 + m*4.75
xm = 4.75m - 8
where m is an integer in the interval {{{0 < m < 4}}}
When I write xm, I really mean "x subscript m" or {{{x[m]}}}


And
deltaY = (1/n)*|d-b|
deltaY = (1/4)*|7-1|
deltaY = 6/4
deltaY = 1.5
This is the increment we move along the vertical axis when determining each cutoff point.
Furthermore,
ym = b + m*deltaY
ym = 1 + m*1.5
ym = 1.5m + 1


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We have this so far
xm = 4.75m - 8
ym = 1.5m + 1


So,
(xm,ym) = (4.75m-8, 1.5m+1)
where m is an integer such that 0 < m < 4
This represents the general form of our cutoff points.


We'll plug in integer values from m = 1 to m = 3.


For example, if m = 1, then...
xm = 4.75m-8
x1 = 4.75*1-8
x1 = 4.75-8
x1 = -3.25
ym = 1.5m+1
y1 = 1.5*1+1
y1 = 1.5+1
y1 = 2.5
This gives one cutoff point at (x1,y1) = <font color=red>(-3.25,2.5)</font>
This is <font color=red>point B</font> shown below.


If m = 2, then,
xm = 4.75m-8
x2 = 4.75*2-8
x2 =9.5-8
x2 = 1.5
ym = 1.5m+1
y2 = 1.5*2+1
y2 = 3+1
y2 = 4
This gives another cutoff point at (x2,y2) = <font color=red>(1.5,4)</font>
This is <font color=red>point C</font> shown below.


Here's a table of values to keep things organized. 
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"><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></table>


Here's what the graph looks like
{{{drawing(500,500,-10,15,-3,8,
graph(500,500,-10,15,-3,8,0),
line(-8,1,11,7),
circle(-8,1,0),
circle(-8,1,0.1),
circle(-8,1,0.2),
red(circle(-3.25,2.5,0)),
red(circle(-3.25,2.5,0.1)),
red(circle(-3.25,2.5,0.2)),
red(circle(1.5,4,0)),
red(circle(1.5,4,0.1)),
red(circle(1.5,4,0.2)),
red(circle(6.25,5.5,0)),
red(circle(6.25,5.5,0.1)),
red(circle(6.25,5.5,0.2)),
circle(11,7,0),
circle(11,7,0.1),
circle(11,7,0.2),
locate(-8,1-0.25,"A"),
red(locate(-3.25,2.5-0.25,"B")),
red(locate(1.5,4-0.25,"C")),
red(locate(6.25,5.5-0.25,"D")),
locate(11,7-0.25,"E")
)}}}
The answer points are marked in red.


A = (-8,1) is one endpoint
<font color=red>B = (-3.25,2.5)
C = (1.5,4)
D = (6.25,5.5)</font>
E = (11,7) is the other endpoint


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To confirm the answers, use the distance formula to show the following
AE = sqrt(397)
AB = sqrt(24.8125)
BC = sqrt(24.8125)
CD = sqrt(24.8125)
DE = sqrt(24.8125)
This will show that AB=BC=CD=DE and that segment AE has been cut into four equal pieces. 
I'll let the student do these steps for the confirmation.


Note:
AE = 4*AB
sqrt(397) = 4*sqrt(24.8125)


Another way to confirm the answers is to use something like GeoGebra. 


Yet another confirmation method is to use the midpoint formula on endpoints A and E, and it will pinpoint where C is located. 
Then use the midpoint formula on A and C to find where B is located.
Finally, use the midpoint formula on C and E to find where D is located.
This repeated midpoint trick only works when there are 2^k equal pieces, where k is some positive integer.
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