Collinear Points are sets of three or more than three points that lie in a straight line. In simple words, if three or more points are collinear, they can be connected with a straight line without any change in slope.
In this article, we will discuss the concept of collinear points, collinear point definition, collinear point meaning, and properties. We will also know how to determine the three points collinearity by different methods. Further, we will also solve various examples and provide practice questions for a better understanding of the concept of this article.
What are Collinear Points?
Collinear Points are sets of points that all are on the same straight line. These points can lie on different planes but not on different lines. By using the sets of three collinear points, we can draw only one straight line. A straight line can always be drawn by using two points.
So, here we can say that two points are always collinear points. Uncertainty comes when it is more than two points; hence we check three points of collinearity. The word ‘Collinear’ breaks down into the term “co-” which means togetherness and “linear” indicates on the line.
Collinear Points Geometry Definition
The points that come between two other points on the same straight line are called collinear points.
In geometry or maths, collinear points are three or more than three points that are displayed on the same straight line. These points are aligned in a row.
Real-life examples include small and long arrows of the clock from collinear points, when the clock strikes six, students standing in a straight line in the assembly hall, parking of trucks in a straight line on the side of the road, food items on a single straight skewer, points along the corner made by adjoining two walls, car parked in a row.
Read more about Geometry.
Collinear Points in Mathematics
In Mathematics, Collinear Points are the points that are positioned on the same straight line or in a single line. To understand this concept, let’s consider we have three points X, Y, and Z. If X, Y, and Z are collinear, it means we can draw a straight line that passes through X, Y, and Z points and all these points will lie on the same line.
Collinear Meaning
Collinear means to the points that hang on the same line in a linear fashion or we can say that three or more objects are placed in a straightened way, which means in a row, that objects are called lying in collinear.
Non Collinear Points
Non-collinear points are sets of points that do not lie on a single straight line. No single straight line can be drawn by using these points. For example different peak points in the Himalayan range.
Properties of Non-Collinear Points
Some of the key properties of non-collinear points are:
- Points that are positioned on a single line called collinear and not positioned in a collinear way are known as non-linear points.
- There are three most often used ways to determine whether points are collinear or not.
- If three points are collinear, the slopes structured by any two points are equal to the slope structured by the other two.
- The value of the area of the triangle structured by any three collinear points will always be zero.
How to Find if the Points are Collinear?
There are three basic ways of finding if three points are collinear or not. There are various ways to estimate whether the three points are parallel or not but we will discuss the three most often used formulae to determine whether three points are collinear or not. The following formulas for collinear points are-
- Distance Formula Method
- Slope Formula Method
- Area of Triangle Formula Method
Distance Formula Method
We apply the distance formula to determine the difference between three points whose coordinates are known and lie on the same line. This is the very least used method to calculate the collinearity of given points.
Let’s consider the points A, B and C with their coordinates (x1, y1), (x2, y2) and (x3, y3) respectively are three collinear points, then,
Distance from A to B + Distance from B to C = Distance from A to C
AB + BC = AC
√{(x2 -x1 )2 + (y2 -y1 )2 + √ (x3 -x2 )2 + (y3 – y2 )2} = √{(x3 -x1 )2+(y3 -y1)2}
Hence, A, B and C points are collinear.
Learn more about Distance Formula.
Slope Formula Method
Slope method is used to determine whether all three points are definitely on the same line or not by using their coordinates. In simple words, these three points are collinear if their all slopes are equal. Slope method are most accurate method to determine the collinear points. Three points A, B, and C, will only be collinear if
Slope of line AB = Slope of line BC = Slope of line CA.
Let’s consider the points A, B and C with their coordinates (x1, y1), (x2, y2) and (x3, y3) respectively lying on the same line.
Hence, slope of line AB = slope of line BC= slope of line CA
mAB = mBC = mCA
OR
(y2 -y1 )/(x2 -x1 ) = (y3 -y2 )/(x3 -x2 ) = (y3 -y1 )/(x3 -x1 )
Here, m denotes the slope of the line.
Read more about Slope of the Line.
Area of Triangle Method
Area of Triangle method is used to determine the collinearity of three points if their area is equal to zero. In simple words, the triangle created by three points whose coordinates are known will only be collinear if it does not contain any area.
Let’s consider the points A, B and C of a triangle with their coordinates (x1, y1), (x2, y2) and (x3, y3) respectively :
Area of Triangle (⧍ABC) = 0
OR
1/2[x1(y2 – y3 ) + x2 (y3 – y1 ) + x3 (y1 – y2 ) = 0
Also, Check
Solved Examples on Collinear Points
Example 1: Show that points A(5, -2), B(4, -1) and C(1, 2) are collinear points using the Distance Method.
Solution:
The points A, B and C are collinear,
if, (Distance from A to B) + (Distance from B to C) = (Distance from C to A)
By using Distance method, we can determine the distance between two points.
Here x1= 5, y1= -2, x2 = 4, y2= -1, x3 = 1, y3= 2
Distance between AB = √{(x2 -x1 )2 + (y2 -y1 )2 }
⇒ Distance between AB = √{(4 -5)2 + (-1 -(-2))2}
⇒ Distance between AB = √2
Distance between BC = √{(x3 -x2 )2 + (y3 – y2 )2}
⇒ Distance between BC = √{(1 -4 )2 + (2 -(-1))2}
⇒ Distance between BC = √18
⇒ Distance between BC = 3√2
Distance between AC = √{(x3 -x1 )2+(y3 -y1)2}
⇒ Distance between AC = √{(1 -5 )2+(2 -(-2))2}
⇒ Distance between AC = √32
⇒ Distance between AC = 4√2
Therefore, AB + BC = √2 + 3√2 = 4√2
Thus, AB + BC = AC
Hence, all given three points A, B, and C are collinear.
Example 2: Find A(2, 3), B(4, 7) and C(6, 11) are collinear points using slope method.
Solution:
Given the coordinates of point A, B and C are (2 ,3 ), (4 ,7) and (6 ,11 )
The points A, B and C are collinear,
if, slope of line AB = slope of line BC= slope of line CA
⇒ mAB =mBC = mCA
(y2 -y1 )/(x2 -x1 ) = (y3 -y2 )/(x3 -x2 )=(y3 -y1 )/(x3 -x1 )
Here, x1= 2 y1= 3, x2= 4, y2= 7, x3= 6, y3= 11
⇒ (7 -3 )/(4 -2 ) = (11 -7 )/(6 -4)=(11 -3 )/(6 -2)
⇒ 4/2 = 4/2 =8/4
⇒ 2 = 2 = 2
Hence, slopes of all three points are equal.
Therefore, the three points A, B and C are collinear.
Example 3: Show that points A(2, 3), B(4, 7), and C(6, 11) are collinear points using the Area of triangle Method.
Solution:
Given points A, B and C of triangle with their coordinates (2,3) (4,7 ) and (6,11 )
Area of triangle (⧍ABC) = 0
⇒ 1/2[x1(y2 -y3 ) + x2 (y3-y1 ) + x3 (y1 -y2 )] = 0
⇒ 1/2[2(7 -11 ) + 4 (11-3 ) + 6 (3 -7)]
⇒ 1/2[-8+32-24] = 0
⇒ Area = 0
Hence, the points A, B and C are collinear.
Example 4: If the collinear points are given (a,0), (0,b) and (1,1). Then find out the value of (1/a + 1/b)2.
Solution:
Given points A, B and C of triangle with their coordinates (a,0) (0,b) and (1,1)
Area of triangle (⧍ABC) = 0
⇒ 1/2[x1(y2 -y3 ) + x2 (y3-y1 ) + x3 (y1 -y2)] = 0
⇒ x1(y2 – y3 ) + x2(y3 – y1 ) + x3(y1 – y2 ) = 0
⇒ a(b-1) + 0(1-0) + 1(0-b) =0
⇒ ab – a – b=0
⇒ ab = a +b
⇒ ab/a+b =1
⇒ 1/a + 1/b = 1
⇒ (1/a + 1/b)2 = 1
Practice Questions on Collinear Points
Q1. What does collinear mean by definition?
Q2. What are collinear points in mathematics?
Q3. What are non-collinear points? Give some real examples of it.
Q4. Give some real examples of collinear points.
Q5. Show that points A(3,7), B(6,5) and C(15-1) are collinear.
Q6. Find if the given points A(0,3), B (1,5) and C (-1,1) are collinear.
Q7. Show that points A (5,2), B (3,-2) and C (8,8) are collinear points by using the Distance Method.
Q8. Show that the points A(2,-1) B(6,4) and C(4,3) are collinear by using the Area triangle method.
Q9. Check that the points A(7, -5), B(9, -3) and C( 13 1) are collinear or not by using slope formula.
Q10. Write down the distance, slope and area of the triangle formula.
FAQs on Collinear Points
1. What is the Meaning of Collinear Points?
When three or more points lie on the same straight line or in a linear way, they are known as collinear points.
2. Can we Draw a Straight Line by using Four Collinear Points?
Yes, we can draw a straight line by using four collinear points as all the collinear points lie on the same line.
3. What is the Meaning of Non-Collinear Points?
Non-collinear points are set of points, when we positioned all on a plane, does not show single straight line are called non-collinear points.
4. How many Methods can we Use to Calculate the Collinear Points?
There are three most often used methods to find out that three points are collinear or not
- Distance Method
- Slope Method
- Area of Triangle Method
5. Do Collinear Points have to be Equal?
Yes, by definition, sets of point are collinear if they all positioned in a same straight line.
6. Which Method is Most Appropriate Method to Determine the Collinear Points?
Slope method is the most appropriate method to determine Collinear Points
7. What is the Slope Method to check Collinearity?
Slope method are used to determine that all the points are are collinear or not by using their coordinates.In simple words these three points are collinear if their all slopes are equal.
8. Can more than Three Points be Collinear?
Yes, any number of points can be collinear if they all lie on the same straight line.
9. How many Points are needed to determine a Line in a Plane?
There should me minimum two points and maximum more than two.
10. How can we Find the Slope of a Line?
If two points A(x1 ,y1) and B(x2, y2) are given with their coordinates in a plane, then its slope would be m = (y2 -y1 )/(x2 -x1 ).
11. Can we Show all the Collinear Points on a Line?
Yes, we can show all the collinear points on a line because it is the definition of the collinear points.
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