We all have struggled with math’s once in our life. Even if you are good at the subject, you will come across various questions either at your school level or any entrance examination where you may have needed some help to understand the concept. For the students preparing for their upcoming examination for math’s, Geometry is a very important topic. This topic usually takes up a big chunk in the total syllabus, but geometry can be quite scoring once grasped.
A student should have a better understanding of Geometry because it is not only limited to your subject but also applies to real-life situations. Geometry is all about shapes, which include lines, curves, rays, etc. To understand Collinear Points in Geometry first, we need to understand the basics of lines and angles. So, without wasting any time, let’s get started.
What Are Lines and Angles?
Lines: Lines are the figures that are made up of infinite points extending towards both directions. Lines are usually straight and have considerable width and depth. Lines can be perpendicular, parallel, intersecting, transversal, etc. Lines are further categorized as line segments and rays.
Angle: An angle is a figure which is formed by two rays that are meeting at a common end-point. The relation between two angles can either be Supplementary, Complementary, Adjacent and Vertically Opposite. An Angle can be categorized as Acute Angle(<90 degrees), Right Angle(=90 degrees), Obtuse Angle ( >90 degrees), straight Angle (=180 degrees).
Understanding the concept better
Line Segment: A part of a line that is bounded by two endpoints. It is the shortest distance between two points and has a fixed length.
RAY: A ray in geometry can be defined as a line with a fixed starting point and no endpoint. It can infinitely extend towards a single direction or may pass through multiple points.
Point: A point indicates a position or location in space. A point has no shape or size and is represented by a small dot. Example: “. A” or “Point A.”
Plane: Plane is a flat two-dimensional surface which may contain points, lines, segment, etc
Types Of Lines
- Perpendicular Lines: In a branch of Geometry, Perpendicular lines are defined as when two lines that meet or intersect each other at right angles (90 degrees)
- Parallel Lines: The two lines are said to be parallel when they do not meet at any point or intersect each other
Transversal Line: A Transversal line is a line that cuts or intersects two lines at distinct points.
Properties of Lines
- Collinear points: Collinear points are sets of three or more than three points, which lie on the same plane.
- The points that don’t lie on the same plane are called noncollinear points.
Types of Angles
An Angle is measured in degrees and can range from a value “0” to “360.”
Based on their measurements, they can be divided into:
- Acute Angle: The measurement of the angle is between 0 to 90 degrees.
- Right Angle: The measurement of the angle is 90 degrees.
- Obtuse Angle: The measurement of the angle is between 90 to 180 degrees.
- Reflex Angle: The measurement of the angle is between 180 to 360 degrees.
- Complementary Angle: Two angles which sum up to form 90 degrees.
- Supplementary Angle: Two angles that sum up to form 180 degrees.
- Adjacent Angle: Two angles that have a common side and vertex are called adjacent angles.
- Vertically Opposite Angles: Two angles that are formed opposite to each other when two lines intersect at a common vertex.
Properties of Angle
- When two rays emerge from a common point, then an angle is formed. The common point is called a vertex, and the two rays forming are called its arms or sides.
- An angle greater than 180 but less than 360 degrees is known as Reflex angle.
- If two adjacent angles sum up to form 180 degrees, they form a linear pair.
- When two lines intersect each other to form opposite pairs, they are called vertically opposite angles.
Collinear Points and Non-collinear points
- Collinear Points: Collinear points are the points that lie on the same plane either in the same line, apart, or from a ray. Co: Together, and linear: line.
- Non-collinear Points: These are the set of points that do not lie on the same line.
- Collinearity in Geometry: Collinearity in Geometry is the property of the points lying on a single line. In Euclidean Geometry, this relation is visualized by the points lying in a row or a straight line. However, in geometry, a line is typically a primitive (object type), so such visualizations will not be considered appropriate. Geometry offers a model of how these points, lines, and objects are in relation to each other and how collinearity can be interpreted between them. In Spherical Geometry, where a model of a circle represents lines, the sets of Collinear points will lie on the same great circle and do not lie on a straight line.
Application of Collinear points and Non-collinear points in real-life situations
- Collinear points: Suppose in a series of individual items located in a straight line. For example, let’s take a row of eggs in a carton when each egg placed in a single line or few numbers of students seated in a long classroom table is called Collinear points. The objects should be placed in such a way that you should be able to draw a straight line through them.
- You can also see Collinear points in various food items available worldwide such as Dango in Japan, which are sweet little dumplings arranged in three or five skewers, or a South African dish called Sosatie, which is little cubes of Lamb or Mutton with dried apricots, red onions, mixed peppers, etc. arranged on a skewer.
- Non-collinear Points: These are points that do not lie on the same plane. Suppose you have multiple sushi rolls placed on different skewers. To have collinearity, all the points should lie on the same line. Since they are all located on different skewers, a line cannot be formed using all the points together, and hence, it will form Non-collinear points. Imagine a picture of a right angle with two different points labeled L and R. If L is forming hypotenuse and R is forming the base then, both are Non-collinear to each other.
Formulas used for Collinear Points
You can find Collinear points using two methods:
- Slope Formula
- Area of Triangle
Slope Formula: The Slope formula is measured as the ratio of the change in the Y-axis to the change in the X-axis. The straight-line slope will describe the steepness of the line’s angle from the horizontal, whether it is rising or falling. If it’s neither rising nor falling, then the value of the slope will be zero.
The formula to calculate slope is,
m= y2-y1/x2-x1
Where m= slope of the line
Let us solve a sample question.
Question: Prove that the three points R(2,4) S ( 4,6) & T( 6,8) are collinear
Ans: If R(2,4) S( 4,6) and T( 6,8) are collinear, then the slopes of any of the two pairs will be equal.
Using Slope formula:
Slope of RS= (6-4)/ (4-2)= 1
Slope of ST =(8-6)/ (6-4)= 1
Slope of RT =(8-4)/ (6-2)= 1
Since slopes of any two pairs out of the given 3 pairs are equal, then it proves R, S, and T are Collinear.
Area of a triangle: If an area of a triangle is formed by three points is zero, they are collinear. When three sets of points are collinear, then they cannot form a triangle.
Area of Triangle is measured by,
½ { x1-x2 x2-x3 }
Y1-y2 y2-y3
Let’s solve an equation using the above-given coordinates (R, S, and T) using the area of triangle formula to check if the answer is Zero.
½ [ 2-4 4-6] = ½ [-2 – 2] = ½ (4-4)=0
4-6 6-8. -2 -2
Here, the result of the formula area of the triangle is zero. So R(2,4), S(4,6), and T(6,8) are collinear points.
Example of Non-collinear points
Question: What is the least number for a Non-collinear point required to determine a plane is
1. 1
2. 2
3. 3
4. Infinite
Answer: If points do not lie on the same plane, it is known as Non-collinear points. If on a plane, we plot two points, it determines the equation of that line. Since a minimum of two points determines the line, it will become a plane if one more point is added. Hence, a minimum of 3 points are required to determine a plane so, the correct answer would be 3.
Conclusion
We hope we have been able to make you understand the basics of Collinear points and Non-collinear points and their relation with Geometry. To have detailed knowledge about this topic, please refer to the extensive collection of notes on various math topics from Cuemath. Apart from comprehensive explanations and notes, Geometry made easy with Cuemath, as they also have the latest questions to practice that are important for your upcoming examination.
Know more about Cuemath through their free trial class and site as they keep on building your concepts and focus better on having a stronghold on the subject of Math without any need for external help.
