Find all points of division.

Question

The line segment joining (-5, -3) and (3,4) is to be divided into five equal parts.

Answer 1

Each point in the Cartesian plane is described by (x,y), where x represents the position on the x-axis (the horizontally-directed line) and y represents the position on the y-axis (the vertically-directed line).

The line segment between two points, A (x1,y1) and B (x2,y2), is the hypotenuse of a triangle that line segment forms with a third point, either C (x1,y2) or D (x2,y1).  The Pythagorean Theorem allows us to calculate the length of the line segment (the segment AB).

 

This means that if distances AB, AC, and BC are known, that any point E on the segment forms a similar triangle (note: Figures that are the same shape but not necessarily the same size are called similarfigures).  We use this information when we calculate the midpoint of segment AB to be the point with coordinates (midpoint-of-x-values, mid-point-of-y-values).

 

So, we simply need to divide x-values from -5 to 3 into five equal parts and to divide y-values from -3 to 4 into five equal parts.

     (3-(-5))/5 = 8/5 for each x-value increase

     (4-(-3))/5 = 7/5 for each y-value increase

 

Here are the points (let’s do everything in improper fractions, but you can convert them):

     x-value       y-value

       -25/5          -15/5      start at point (-5,-3)

       -17/5           -8/5

         -9/5           -1/5

         -1/5            6/5

          7/5           13/5

        15/5           20/5        end at point (3,4)

(note: with endpoints, five equal parts is 6 points on the line segment)

 

We make it to point (3,4) !!

Answer 2

There is a formula for finding division points on a live segment.

If you divide a line segment into a ratio m:n ...

 

To find a point of division between (x₁,y₁) and (x₂,y₂) in ration m:n

The point of division is:  [(mx₂ + nx₁)/(m+n), (my₂+ny₁)/(m+n)]

 

In this case we want to divide the segment into 5 equal parts so

there will be 4 points of division.

 

For the first division point, we will take as being the closest to the

bottom point of our segment at (-5,-3) ...

Since we are dividing into 5 equal lengths the ratio from (-5,-3)

to the 1st division point will cause the line to be divided in a ratio

of 1:4   so  m=1, n=4   (x₁,y₁) = (-5,-3),  (x₂,y₂) = (3,4)

 

The x coord for 1st point of division is  (1(3)+4(-5))/(1+4) = -17/5 = -3 2/5

The y coord for 1st point of division is  (1(4)+4(-3))/(1+4) = -8/5 = -1 3/5

The first point of division is (-3 2/5, -1 3/5)

 

Still using (x₁,y₁) = (-5,-3) and (x₂,y₂) = (3,4) for the 2nd point of

division our new ratio will now be 2:3  so  m = 2,  n = 3

 

The x coord for 2nd division point is (2(3)+3(-5))/(2+3) = -9/5 = -1 4/5

The y coord for 2nd division point is (2(4)+3(-3))/(2+3) = -1/5

The 2nd point of division will be (-1 4/5, -1/5)

 

Continuing in this vein the ration for the 3rd point of division is 3:2

Still using the formula [(mx₂ + nx₁)/(m+n), (my₂+ny₁)/(m+n)]

If we continue using (-5,-3) as (x₁,y₁) and ((3,4) as (x₂,y₂) with m =3, n=2

 

For the fourth point of division the ratio m:n would become 4:1

 

I leave the finding of the other 2 points of division to you.  Good luck and

I hope this helps.

 

Andrew

 

BTW:  Let's verify that our 2nd point of division (-1 4/5 , -1/5) is actually

on our original line segment:

We have points (-5,-3), (3,4)

The slope is rise/run = (4-(-3))/(3-(-5)) = 7/8

our line segment looks like y = (7/8)x + b

Plugging in (3,4) for (x,y) we can find b

4 = (7/8)(3) + b

4 = 21/8 + b

b = 4-(21/8) = 11/8

The line segment equation is y = (7/8)x + 11/8

 

We need to verify that (-1 4/5, -1/5) is on that line

or (-9/5, -1/5)

-1/5 = (7/8)(-9/5) + 11/8

-1/5 = -63/40 + 11/8

-1/5 = -63/40 + 55/40

-1/5 = -8/40

-1/5 = -1/5

The point verifies as being on the line segment so our answer

appears to be correct. 

 

Answer 3

Draw the points on a coordinate system and connect them.  Then draw some points on the line so that you have 5 equal divisions.  It will help give you a visual of what I am going to explain.

 

The given points form a single line.  Since we want to divide the line into 5 equal parts, we need to find 4 points that are equidistant from each other.  Basically, all the points on this line are equidistant from each other.

 

 

Each point on the line is 1/5 the distance of a reference point. We don't know what that distance is.  So we use the distance formula using the two endpoint to find that distance.

 

d = √[(x₁ - x₂) + (y₁ - y₂)²]

 

d = √[(-5 - 3)² + (-3 - 4)²]

 

d = √(64 + 1)

 

d = √65

 

 

The slope is also important because it keeps the points on that line.  Find the slope.

 

Slope = (4 - (-3)) / (3 - (-5))

Slope = 7 / 8

 

 

The distance of this line is √65.

The slope is 7/8. 

 

Keep the slope in mind when finding the division points.  The slope keeps the points on that line.

 

The distance between each point on this line is 1/5 the distance of this line.  We will use a starting point to find each point of division.  Let that starting point be (-5, -3) and work our way all the way to (3, 4). 

 

Let A be the first point of division.  This point is closest to (-5, -3).  This make D the last point of division and closest to (3, 4).

 

 

First point of division:

 

√(65) / 5 = √[(-5 - x_A)² + (-3 - y_A)²]

 

 

Second point of division:

 

√(65) / 5 = √[(x_A - x_B)² + (y_A - y_B)²]

 

 

Third point of division:

 

√(65) / 5 = √[(x_B - x_C)² + (y_B - y_C)²]

 

 

Continue this pattern to find the points.