Determining whether points are collinear

Question

$(1,1)(3,9)(6,21)$

The way I figured that this should be solved is by finding the slope of: $(1,1)(3,9)$

Then, $(3,9)(6,21)$

Finally $(1,1)(6,21)$

Which are 4, 4,and 4 respectively. So I assume that they are collinear.

Am I correct? And if not, please provide me with an explanation as to what needs to be done to find the answer rather than a direct answer.

How about some linear algebra and a calculating some determinant? — Lolman, Aug 31 '14 at 01:57
To be correct, you need to give an answer to your question: "Are those three points collinear?" — JimmyK4542, Aug 31 '14 at 01:58
Well they all have the same slope so I assume that this means that they are collinear..? — Cherry_Developer, Aug 31 '14 at 01:59
Yes, if AB and BC have same slopes then the points A, B, C are collinear. Finding the third slope AC is redundant. — AgentS, Aug 31 '14 at 02:02
@Cherry_Developer, I am very sorry for the incorrect solution, you need NOT find $y$ intercept. First you find slope of $AB$, next you find slope of $BC$ and we see that slope of $AB$=slope of $BC$ and since $B$ is common to both the line segments, it proves that the three points are collinear — Vikram, Aug 31 '14 at 08:05

score 1 · Answer 1 · answered Oct 10 '15 at 10:27

1

What you checked is ok; in plane geometry if any two slopes from the point pairs are necessarily same, it is sufficient to conclude collinearity.

answered Oct 10 '15 at 10:27

Narasimham

40,495

score 0 · Answer 2 · answered May 16 '15 at 11:42

As you're given three points yes it's correct.
There are more ways to find co-linearity:
Determinant method: If $$\det\left| \begin{array}{ccc}1 & 1 & 1 \\ 3 & 9 & 1 \\ 6 & 21 & 1 \end{array}\right|=0$$ It is colinear.
Using distance formula: if your points are A,B and C then, $$AB^2+BC^2=AC^2\ \ or\ \ AC^2+BC^2=AB^2\ \ or\ \ AB^2+AC^2=BC^2$$ means it is colinear

score 0 · Answer 3 · answered Oct 10 '15 at 10:07

0

Yes, you are correct. It is sufficient to calculate only two slopes to decide that three points are collinear.

answered Oct 10 '15 at 10:07

miracle173

11,049

Determining whether points are collinear

3 Answers3