Graph the segment on a coordinate plane. How many steps up and to the right do you need to take to get from point A to point B? How does this number of steps differ if you want to get to the midpoint of the segment?
(- 3+12/n, 5 10n ), (- 3+24/n, 5 20n ),
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Practice makes perfect
Let's plot the given points, A and B, and draw the segment that connects them on a coordinate plane.
To get from A to B we have to move 12 steps to the right and then 10 steps up. If we wanted to find the coordinates of the point that divides the segment into 2 pieces, we could use the Midpoint Formula. The midpoint occurs at half the number of steps we need to move from A to B.
12/2=6 and 10/2=5
To
We have to move 6 steps to the right and then 5 steps up. If we want to get to the midpoint from A.
Therefore, the midpoint of this segment is (3,10).
To find the coordinates of the points that divide AB into n segments, we can follow the same process. This time, we will divide 12 and 10 by n first.
Change inx: 12/n [1em]
Change iny: 10/n
Now, we can find the coordinates of the first point by going 12n steps to the right of A and then 10n steps up. This means that we add 12n and 10n to the x - and y -coordinate of point A, respectively.
Point A
(x+12/n,y+10/n)
New point
(-3,5)
(-3+ 12/n, 5+ 10/n)
(- 3+12/n, 5 10n)
To find each new point, we need to repeat this process until we have the n-1 points that divide AB into n segments.
Point
(x+12/n,y+10/n)
New point
(- 3+12/n,5 10n)
(- 3+12/n+12/n, 5 10n+10/n)
( - 3+24/n , 5 20n)
( - 3+24/n ,5 20n)
( - 3+24/n+12/n, 5 20n+10/n)
(- 3 +36/n,5 30n)
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Notice that the coordinates of these points follow a pattern.
(-3+ 12/np, 5+ 10/np)
In the above pattern, the variable p is the number of the point along the segment after A.
