Introduction

For any two distinct points, there is always one (and only one) line that passes through them.

For example, the line passing through the points P and Q is \overset{\longleftrightarrow}{PQ}, while the line that passes through A and B is \overset{\longleftrightarrow}{AB}\mathbin{:}



However, given three or more points, there is not always a line that passes through all of them.

For example, for the three points F, G, and H below, it is impossible to draw a single straight line that passes through all of them.



However, it is sometimes possible to draw a line through three (or more) points. In such cases, we say that the points are collinear.

For example, the points M, N, and O shown below are collinear since they all lie on the same line.



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Example: Identifying Collinear Points on a Grid

Which of the diagrams below shows three collinear points?

EXPLANATION

Three (or more) points are collinear if a straight line passes through all points.

With that in mind, let's examine our diagrams in turn.

  • Diagram I shows three points that are not collinear. We can't draw a straight line that passes through all three points.
  • Diagram II shows three collinear points. There is a straight line that passes through all three of these points.
  • Diagram III shows three points that are not collinear. We can't draw a straight line that passes through all three points.

Therefore, the correct answer is "II only."

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Practice: Identifying Collinear Points on a Grid

Which of the diagrams above shows three collinear points?

a
 III only
b
 II only
c
 I only
d
 I and II only
e
 II and III only
Practice: Identifying Collinear Points on a Grid

Which of the diagrams above shows three collinear points?

a
 II only
b
 I only
c
 I and III only
d
 III only
e
 II and III only
Example: Determining Whether Points on Intersecting Lines Are Collinear


Given the picture shown above, which of the following statements are true?

  1. The points K , N , and Z are collinear
  2. The points K , P , and J are collinear
EXPLANATION
  • Statement I is false. We can't draw a straight line that passes through these three points.



  • Statement II is true. There is a straight line that passes through these three points.



Therefore, the correct answer is "II only."

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Practice: Determining Whether Points on Intersecting Lines Are Collinear

Consider the figure shown above. Which of the following triplets of points are collinear?

  1. $M, N, R$
  2. $P, S, T$
  3. $O, P, S$
a
 I only
b
 I and III only
c
 I and II only
d
 II and III only
e
 III only
Practice: Determining Whether Points on Intersecting Lines Are Collinear

Given the picture shown above, which of the following statements are true?

  1. $E$, $G$ and $H$ are collinear
  2. $A$, $B$ and $G$ are not collinear
  3. $B$, $F$ and $H$ are collinear
a
 III only
b
 II only
c
 I only
d
 I and III only
e
 I and II only
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