SOLUTION: solve by addition of line segments show whether the points a(-3,0), b(-1,-1) and c(5,-4) lie on a straight line.

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Question 1203418: solve by addition of line segments show whether the points a(-3,0), b(-1,-1) and c(5,-4) lie on a straight line.




Found 7 solutions by josgarithmetic, Theo, greenestamps, Edwin McCravy, ikleyn, math_tutor2020, mccravyedwin:
Answer by josgarithmetic(39620)   (Show Source): You can put this solution on YOUR website!
"addition of line segments" ?
In what way?
Do you mean Use Of Distance Formula?

To know if the three points collinear, find slopes of every pair of them. All same? Line.
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
your 3 points are:
a = (-3,0)
b = (-1,-1)
c = (5,-4)

if they are on the same straight line, then they should be able to form an equation with the same slope and same y-interept.

straight line formula is y = mx + b
m is the slope
b is the y-intercept.

you can find the straight line equation from any two of the ponts.
then you can determine if the third point is on the same line by using the equation.

i chose points (-3,0) and (5,-4)
slope is (y2-y1) / (x2-x1)
assigned (-3,0) to (x1,y1)
assigned (5,-4) to (x2,y2)

(y2-y1)/(x2-x1) = (-4-0) / (5--3) = -4/8 = -1/2.

y = mx + b becomes y = -1/2 * x + b

to find the y-intercept, replace x and y with values from one of the points.
i chose (5,-4).
at that point, x = 5 and y = -4
y = -1/2 * x + b becomes -4 = -1/2 * 5 + b
solve for b to get b = -4 + 1/2 * 5 = -4 + 2.5 = -1.5
y-intercept is -1.5 and equation becomes y = -1/2 * x - .5

your third point should be (-1,-1).
to prove that is on the line, replace x in the equation with -1 and solve for y.
you get y = -1/2 * -1 - 1.5 which is equal to 1/2 - 1.5 which is equal to -1.
this means the point (-1,-1) is also on the same line.

here is a graph of the equation y = -1/2 * x - 1.5.
it shows that all 3 points are on the same line.





Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


I'm not sure why the other tutors have a problem with "addition of line segments".

The three points are collinear if (and only if) the length of AC is the sum of the lengths of AB and BC.

AB:

BC:

AC:



The three points are collinear.
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
The first two tutors did it by slopes and showing they are points on the same
line.  Those are valid methods, but the method asked for here is how to do it
by addition of line segments.



We use the distance formula to find the length of the line segment from a(-3,0)
to b(-1,-1)



Now we use the distance formula also to find the length of the line segment
from the point b(-1,-1) to c(5,-4)



Now we find the length of the line segment from a(-3,0) to c(5,-4) to see if the
sum of those two segments equals to this total length.



And we see that the sum of the two shorter line segments equal to the whole
longest line segment:



That proves that all three points lie on the same line.

Edwin
Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!
.

Comparing with using slopes, using distances for this problem is very ineffective method,

                - - - since it requires MUCH MORE computations - - - .


Therefore, using the slopes method is traditionally considered as a standard method
solving such problems, while this instruction to use the distances (so called "addition of line segments")
is puzzling.

It teaches to enter a house through a window instead of doing it through a door.


////////////////////


Edwin, I got your message as a comment.

I always read the input attentively - - - you may learn it observing my posts at the forum for more than 12 years.

What I write, does not have a goal to surprise you or refute your solutions.

What I write, has the only goal to teach the visitors in a way as it should be done
(to that extent as I can do it).

And I am proud that in this year  2023,  from  Jan.1  till the end of  August,
there are just about  235000  clicks to all my about  1150  lessons
at this site,  or about  ~1000  clicks per day.
It means that  THOUSANDS  real people in the world learn from my lessons.

In this calculations, I did not count those visitors who learn from my everyday posts . . .
This number can be roughly estimated as (10 posts per day) x (240 days in the year 2023 from Jan till Aug) = 2400,
which should be taken with a multiplicity coefficient, because many posts/solutions are read by several people.

...............................


Edwin, is it really true that in schools, colleges and/or universities
they teach to solve such problems by this method?

It is really unexpected thing to me.

Never could even imagine it . . .


If they do teach it, then it can be done under one mandatory condition - they always should say explicitly
that this method for these problems is computationally ineffective - exactly what I said in my post.

Otherwise, their teaching is one-sided, crooked and incomplete.

It is the real and exact/precise meaning of my post.



Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

There are great solutions by the other tutors.

Here's another similar question to help give more practice
https://www.algebra.com/algebra/homework/Graphs/Graphs.faq.question.1203348.html
Answer by mccravyedwin(408)   (Show Source): You can put this solution on YOUR website!

Although Ikleyn is correct that using slopes is the quickest 
way to show that points are collinear, it is ABSOLUTELY WRONG  
to do this problem using slopes, because of the words

            "solve by addition of line segments"

are clearly stated in the instructions.  Following directions
is absolutely necessary.  

When I taught this I always taught both methods for instructional
purposes. 

Edwin
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