Collinear Points
Collinear points are important in geometry. They help us understand how points lie on a straight line. You will often see collinear points in math problems. They are used in line drawings, shapes, and graphs.
In this blog, we will explain what collinear points are. We will give simple examples. You will learn how to prove if points are collinear. We will also show the difference between collinear and non-collinear points. Let’s begin with the meaning of collinear points.
1.0Definition of Collinear Points
The definition of collinear points is simple. Collinear points are three or more points that lie on the same straight line. The word “collinear” comes from the Latin word collinearis, where “col” means “together” and “linearis” means “in line.”
So, if points A, B, and C lie on the same line, then they are collinear.
This concept is foundational in geometry and is used in many constructions and proofs.
2.0Visual Representation
To understand collinearity better, look at this visual example:
Here, points A, B, and C lie on the same straight line. Hence, they are collinear.
Now consider:
In this case, point C does not lie on the line formed by points A and B. So, points A, B, and C are not collinear.
3.0Examples of Collinear Points
Here are some examples of collinear points in different contexts:
Example 1: Geometry Line Segment
Here in this figure, using Euclid’s postulate 1, which states that “A straight line may be drawn from any point to any other point.” Hence, according to this postulate:
If we can draw a straight line from A to C, and B lies between A and C on this straight path, then A, B, and C must lie on the same straight line. Since the same straight line passes through all three points (A, B, and C), by definition, they are collinear.
Example 2: Coordinates on a Plane
Consider points with coordinates:
These points lie on the line y = 2x, so they are collinear.
Prove: If all points satisfy the equation y = 2x, then they lie on that line.
- For A(1, 2):
y = 2x → 2×1= 2 - For B(2, 4):
y = 2x → 2×2 = 4 - For C(3, 6):
y = 2x → 2×3 = 6
Since all three points satisfy the same linear equation, they lie on the same straight line.
Example 3: Real-Life Example
On a straight road, if three street lights are placed one after another in a straight direction, those points (where the lights are placed) are collinear.
4.0Collinear Points in Geometry
Collinear points in geometry help in:
- Constructing straight lines
- Understanding slopes and gradients
- Performing coordinate geometry calculations
- Creating geometric proofs
Collinear points often appear in line segments, triangles, and polygons. For example, in a triangle, the midpoints of the sides can sometimes be collinear, forming what is known as a median line or Euler line (in special cases).
5.0How to Prove Points Are Collinear
There are different methods for how to prove points are collinear, depending on the context.
Using Slope in Coordinate Geometry
If three points A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃) are collinear, then the slope of AB should be equal to the slope of BC.
Formula for slope:
Slope of AB=x2−x1y2−y1 Slope of BC=x3−x2y3−y2
If both slopes are equal, the points are collinear.
Example
Let A(2, 3), B(4, 7), and C(6, 11)
Slope of AB = (7 - 3)/(4 - 2) = 4/2 = 2
Slope of BC = (11 - 7)/(6 - 4) = 4/2 = 2
Since both slopes are the same, the points A, B, and C are collinear.
Using the Area of a Triangle
If three points form a triangle with zero area, then the points are collinear.
Formula for area using coordinates:
Area =21[x1(y2−y3)+x2(y3−y1)+x3(y1−y2)]
If the area = 0, then the points are collinear.
Using Vector Cross Product
If the vectors AB and AC are scalar multiples, then A, B, and C are collinear.
This is often used in 3D geometry.
6.0Difference Between Collinear and Non-Collinear Points
Understanding the difference between collinear and non-collinear points is key in geometry.
7.0Important Properties of Collinear Points
- Any two points are always collinear: Two points can always be connected by a straight line.
- Three or more points need to be tested: They are collinear only if they lie on the same straight line.
- Used in coordinate geometry for calculating equations of lines.
- Slope and area methods are commonly used to prove collinearity.
8.0Applications of Collinear Points
Collinear points are used in:
- Designing straight edges in architecture.
- Creating line graphs in data analysis.
- Navigation and GPS tracking.
- Drawing and modelling in CAD software.
- Proving geometric theorems.
9.0Practice Problems
Problem 1
Are the points P(1, 1), Q(2, 2), and R(3, 3) collinear?
Solution:
Slope of PQ =2−12−1=1
Slope of QR =3−23−2=1
Both slopes are equal.
→ Points are collinear.
Problem 2
Do the points M(0, 1), N(1, 3), O(2, 5) lie on the same line? Prove it using formula for the area of a triangle.
Solution: In the given problem, let
M (0,1) = (x1, y1)
N(1,3) = (x2, y2)
O(2,5) = (x3, y3)
Now using the formula:
Area =21[x1(y2−y3)+x2(y3−y1)+x3(y1−y2)] Area =21[0(3−5)+1(5−1)+2(1−3)] Area =21[0+4+(−4)]=21[0]=0
Hence, the points M, N, and O are collinear, which further indicates that these points lie on the same straight line.
Problem 3
Three points A(1, 2), B(3, 6), and C(x, 10) are collinear. Find the value of x.
Solution: In the question, given that:
A(1, 2)) = (x1, y1)
B(3, 6) = (x2, y2)
C(x, 10) = (x3, y3)
Since all the points are collinear, the area of the triangle with these coordinates will be 0. Using this concept:
Area =21[x1(y2−y3)+x2(y3−y1)+x3(y1−y2)]0=21[1(6−10)+3(10−2)+x(2−6)]0=−4+24−4x4x=20x=5
10.0Conclusion
Collinear points are points that lie on the same straight line. They are easy to understand and very useful in geometry. You can check if points are collinear by using slope or area methods. Collinear points help in drawing lines, solving problems, and proving shapes. Knowing the difference between collinear and non-collinear points makes geometry easier. Keep practising to get better at finding collinear points.