Collinear and Non-Collinear Points
When we place points on a surface, they can either all lie on the same straight line or be scattered. If all points lie on the same straight line, they are called collinear. If they do not, they are called non-collinear.
Think of it this way: if you can place a ruler along three dots and all three touch the edge of the ruler, the points are collinear. If one dot is off the ruler's edge, the points are non-collinear.
Understanding collinear and non-collinear points is important in geometry. You need at least three non-collinear points to form a triangle, and collinear points help us understand lines and segments.
In Class 6, you will learn to identify collinear and non-collinear points, check whether given points are collinear, and understand how these concepts connect to lines and shapes.
What is Collinear and Non-Collinear Points?
Definition:
- Collinear points: Three or more points that lie on the same straight line.
- Non-collinear points: Three or more points that do NOT all lie on the same straight line.
Collinear = "Co" (together) + "Linear" (line)
Points that lie together on a line
Key facts:
- Any two points are always collinear (because exactly one line passes through them).
- We use the term "collinear" mainly for three or more points.
- If points are collinear, one single line passes through ALL of them.
- If points are non-collinear, no single line passes through all of them.
Examples:
| Points | Arrangement | Type |
|---|---|---|
| A, B, C on a straight line | All on one line | Collinear |
| P, Q, R forming a triangle | Not on one line | Non-collinear |
| X, Y on a line | Always on one line | Always collinear (just 2 points) |
Collinear and Non-Collinear Points Formula
How to Check if Points Are Collinear:
Method 1: Visual Check (Drawing Method)
- Plot the points on paper or a graph.
- Try to draw a single straight line through all the points.
- If one line passes through all points → collinear.
- If no single line passes through all → non-collinear.
Method 2: Distance Check
- Measure the distances AB, BC, and AC (for three points A, B, C).
- Check if the longest distance equals the sum of the other two.
- If AB + BC = AC (where C is furthest from A), the points are collinear.
- If AB + BC ≠ AC, they are non-collinear.
Collinearity Test:
If AB + BC = AC, then A, B, C are collinear (B is between A and C).
Important rules:
- Three non-collinear points form a triangle.
- Three non-collinear points determine exactly one plane.
- Through three collinear points, infinitely many planes can pass.
Types and Properties
Situations involving collinear and non-collinear points:
1. Points on a Line
- If points A, B, C, D all lie on line l, they are collinear.
- Any number of points on the same line are collinear.
- Example: Points along the edge of a ruler.
2. Points Forming a Shape
- The three vertices (corners) of a triangle are always non-collinear.
- If three points were collinear, they could not form a triangle — they would just be on a line.
- The four vertices of a rectangle are non-collinear (they cannot all lie on one straight line).
3. Points on a Graph
- If you plot points like (1, 2), (2, 4), (3, 6) on a graph, they all lie on the line y = 2x. They are collinear.
- Points like (0, 0), (1, 1), (2, 3) do NOT lie on a single straight line. They are non-collinear.
4. Lines Through Non-Collinear Points
- Three non-collinear points A, B, C give 3 different lines: line AB, line BC, line AC.
- Four non-collinear points (no 3 collinear) give 6 lines.
- Formula: n non-collinear points give n(n-1)/2 lines.
5. Real-Life Examples
- Collinear: Fence posts along a straight fence. Students standing in a straight line.
- Non-collinear: The three legs of a tripod. Three corners of a room floor (forming a triangle).
Solved Examples
Example 1: Identify Collinear Points
Problem: Points A, B, C are on a straight line. Are they collinear?
Solution:
Steps:
- All three points lie on the same straight line.
- By definition, points on the same line are collinear.
Answer: Yes, A, B, C are collinear.
Example 2: Identify Non-Collinear Points
Problem: Three points P, Q, R form a triangle. Are they collinear or non-collinear?
Solution:
Steps:
- If three points form a triangle, no single straight line passes through all three.
- They are non-collinear.
Answer: P, Q, R are non-collinear.
Example 3: Distance Method Check
Problem: A, B, C are three points. AB = 3 cm, BC = 4 cm, AC = 7 cm. Are they collinear?
Solution:
Given:
- AB = 3 cm, BC = 4 cm, AC = 7 cm
Steps:
- Check: AB + BC = 3 + 4 = 7 cm.
- AC = 7 cm.
- AB + BC = AC. This means B lies between A and C on a straight line.
Answer: Yes, A, B, C are collinear (B is between A and C).
Example 4: Non-Collinear Using Distance
Problem: X, Y, Z are three points. XY = 5 cm, YZ = 4 cm, XZ = 6 cm. Are they collinear?
Solution:
Given:
- XY = 5 cm, YZ = 4 cm, XZ = 6 cm
Steps:
- Check if any distance equals the sum of the other two.
- XY + YZ = 5 + 4 = 9 ≠ 6 (XZ).
- XY + XZ = 5 + 6 = 11 ≠ 4 (YZ).
- YZ + XZ = 4 + 6 = 10 ≠ 5 (XY).
- No combination works.
Answer: No, X, Y, Z are non-collinear. They form a triangle.
Example 5: Lines from Non-Collinear Points
Problem: How many straight lines can be drawn through 3 non-collinear points?
Solution:
Given:
- 3 non-collinear points: A, B, C
Steps:
- Lines possible: AB, BC, AC.
- Each pair of points gives one line.
- Number of lines = 3(3-1)/2 = 3.
Answer: 3 lines can be drawn.
Example 6: Lines from Collinear Points
Problem: How many straight lines can be drawn through 3 collinear points?
Solution:
Steps:
- Collinear points all lie on the same line.
- All three points are on ONE line.
Answer: 1 line passes through all 3 collinear points.
Example 7: Four Points
Problem: Four points A, B, C, D lie on the same line. How many line segments can be named?
Solution:
Given:
- 4 collinear points
Steps:
- Number of segments = n(n-1)/2 = 4 × 3 / 2 = 6.
- Segments: AB, AC, AD, BC, BD, CD.
Answer: 6 line segments: AB, AC, AD, BC, BD, CD.
Example 8: Real-Life Collinear Example
Problem: Students A, B, C are standing in a straight queue. Are they collinear?
Solution:
Steps:
- A straight queue means all students are along one straight line.
- Points along a straight line are collinear.
Answer: Yes, they are collinear.
Example 9: Checking with a Ruler
Problem: Three dots are drawn on a page. How can you check if they are collinear?
Solution:
Steps:
- Place a ruler so that its edge touches the first two dots.
- Check if the third dot also touches the edge.
- If all three dots touch the ruler's edge → collinear.
- If the third dot is off the edge → non-collinear.
Answer: Use a ruler. If all dots align with one straight edge, they are collinear.
Example 10: Minimum Points for a Triangle
Problem: Can 3 collinear points form a triangle?
Solution:
Steps:
- A triangle requires three sides enclosing a region.
- Collinear points are all on the same line — no enclosed region is formed.
- Three non-collinear points are needed for a triangle.
Answer: No. Three collinear points cannot form a triangle. You need non-collinear points.
Real-World Applications
Where collinear and non-collinear points matter:
- Construction: Builders check that bricks in a wall are collinear (in a straight line) using a plumb line or laser level.
- Sports: In cricket, the batsman, stumps, and bowler should be roughly collinear for a straight delivery.
- Navigation: Three landmarks that are collinear can only determine a line, not a position. Navigation requires non-collinear reference points to fix a location (triangulation).
- Drawing shapes: You cannot draw a triangle with collinear points. Every polygon requires non-collinear vertices.
- Surveying: Land surveyors use non-collinear markers to map out areas. Collinear markers only define a boundary line.
- Photography: A photographer lines up subjects (collinear) or arranges them in a group (non-collinear) for different effects.
Key Points to Remember
- Collinear points lie on the same straight line.
- Non-collinear points do NOT all lie on the same straight line.
- Any two points are always collinear.
- The term "collinear" is meaningful for three or more points.
- To test collinearity: check if AB + BC = AC (B between A and C).
- Three non-collinear points form a triangle.
- Three collinear points lie on one line (cannot form a triangle).
- Three non-collinear points determine exactly one plane.
- n non-collinear points (no 3 collinear) give n(n-1)/2 lines.
- You can check collinearity by using a ruler or by measuring distances.
Practice Problems
- Are the three corners of a triangle collinear or non-collinear?
- Points A, B, C have distances AB = 5 cm, BC = 3 cm, AC = 8 cm. Are they collinear?
- Points X, Y, Z have distances XY = 4 cm, YZ = 5 cm, XZ = 7 cm. Are they collinear?
- How many lines pass through 4 non-collinear points (no 3 collinear)?
- Give 3 real-life examples of collinear points.
- Can four points be collinear? Give an example.
- Can you form a triangle with 3 collinear points? Explain.
- Five points are marked on a page. Three of them are collinear. How many distinct lines can be drawn through these points?
Frequently Asked Questions
Q1. What does collinear mean?
Collinear means "lying on the same line." Three or more points are collinear if a single straight line passes through all of them.
Q2. Are two points always collinear?
Yes. Any two points are always collinear because exactly one line passes through them. The concept of collinearity is most useful for three or more points.
Q3. How do you check if three points are collinear?
Method 1: Place a ruler — if all three touch the edge, they are collinear. Method 2: Measure distances AB, BC, AC. If the longest equals the sum of the other two, they are collinear.
Q4. Can non-collinear points form a triangle?
Yes. In fact, you NEED non-collinear points to form a triangle. Three non-collinear points are the vertices of exactly one triangle.
Q5. Can collinear points form a triangle?
No. Collinear points are all on one line, so they cannot form a closed shape. A triangle needs three points that are not on the same line.
Q6. What is the difference between collinear and concurrent?
Collinear describes points that lie on the same line. Concurrent describes lines that pass through the same point. Collinear is about points; concurrent is about lines.
Q7. How many lines can pass through 3 collinear points?
Just 1. Since all three points lie on the same line, only that one line passes through them.
Q8. How many lines can pass through 3 non-collinear points?
3 lines. Each pair of points determines one line: AB, BC, and AC. No single line passes through all three.
