Is my method correct to check if three point lie on a line in euclidean co-ordinates

Question

Question :

Given are three points in euclidean coordinates:

$x_1=(2,2)T$
$x_2=(−8,−8)T$
$x_3=(17,17)T$

Do these three points lie on a line?

My attempt/thinking :

I was given this question. In order to solve this , I calculated the slope of $x_1x_2$ and $x_2x_3$ , to check whether they are not equal or not.

$slope = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-8 - 2} {-8 - 2} = \frac{-10}{-10} = 1$

$slope = \frac{y_3 - y_2}{x_3 - x_2} = \frac{17 + 8}{17 + 8} = \frac{25}{25} = 1$

Since they are equal , they all line on the line.

Issue :

When I submitted the answer , it was flagged as incorrect !
Now I don't know what I did wrong in this case or whether the checking system is wrong.

Answer 1

There are various ways to check whether 3 Points are Co-linear. Your instructor may have given a method which you are supposed to use here , everything else may be automatically marked "wrong" , even though those may be "Correct".

One Potential Issue with your Method may be the wording you used when giving the Answer : "Two lines with Same Slope are Exactly Same"

It must be "Two lines with Same Slope & having a Point in Common are Exactly Same" : Here , the highlighted Part is automatic , because $x_2$ is Common. In Case you missed mentioning this , your Instructor may have marked it that way. You can clarify with your Instructor about this.

Alternatively , your Instructor may be looking for you to figure out the Equation of the line through $x_1$ & $x_2$ , then showing that $x_3$ will satisfy that Equation.

We might also try to figure out the Equation of the line through $x_1$ & $x_2$ , then try to figure out the Equation of the line through $x_2$ & $x_3$ , then showing that we get the Exact Same Equation , indicating that we have the Exact Same line.

While those 3 are the Common methods , we can also check the Distances $x_1x_2$ & $x_2x_3$ & $x_1x_3$. We can also check the area of triangle $x_1x_2x_3$. These other methods are longer & unnecessary here.

You should clarify with your Instructor about which Method to use & what Exactly was wrong in your Method.

Answer 2

Your reasoning is correct but your formulation is dead wrong: you mention starting with three points, called $x_1$, $x_2$ and $x_3$.
Then you start calculating and all of a sudden your $x_1$, $x_2$ and $x_3$ are not entire points anymore, but just the X-coordinates of those mentioned points.

You might better start by saying:

$P_1 = (2,2)T$
$P_2 = (-8, -8)T$
$P_3 = (17, 17)T$

(By the way, I'm just copying your $(2,2)T$ notation but I've never seen a point, being described with the character "T" at the end)

As a result, you might then first say that:

$x_1 = 2$
$y_1 = 2$
$x_2 = -8$
$y_2 = -8$
$x_3 = 17$
$y_3 = 17$
... and continue from there.

Oh, I forgot: you are not calculating two times the same slope, you are calculating $slope_1$ and $slope_2$.