What are Collinear Points in Geometry?

In Euclidean geometry, if two or more than two points lie on a line close to or far from each other, then they are said to be collinear. Collinear points are points that lie on the straight line. The word 'collinear' is the combined word of two Latin names ‘col’ + ‘linear’. ‘prefix ' 'co' and the word 'linear.' 'Co' indicates togetherness, as in coworker or cooperate. 'Linear' refers to a line. You may see many real-life examples of collinearity such as a group of students standing in a straight line, eggs in a carton are kept in a row, next to each other, etc.

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In this article, we will learn the collinear points definition and how to find collinear points.

Collinear Points Definition

In a given plane, three or more points that lie on the same straight line are called collinear points. Two points are always in a straight line. In geometry, the collinearity of a set of points is the property of the points lying on a single line. A set of points with this property is said to be collinear. In general, we can say that points are aligned in a line or a row.

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Consider a straight line in the above cartesian plane formed by x-axis and y-axis.
The three points A (2, 4), B (4, 6), and C (6, 8) are lying on the same straight line L.
These three points are said to be collinear points.

How to Prove if Points are Collinear?

There are two methods to find whether the three points are collinear or not, they are:

One is the slope formula method and
The other is the area of the triangle method.

Slope Formula Method

Three points are collinear, if the slope of any two pairs of points is the same.

With three points R, S, and T, three pairs of points can be formed, they are: RS, ST, and RT.

If the slope of RS = slope of ST = slope of RT, then R, S, and T are collinear points.

Example

Prove that the three points R(2, 4), S (4, 6), and T(6, 8) are collinear.

Solution:

If the three points R (2, 4), S (4, 6), and T (6, 8) are collinear, then the slopes of any two pairs of points will be equal.

Use slope formula to find the slopes of the respective pairs of points:

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 three pairs of points are the same, this proves that R, S, and T are collinear points.

Area of Triangle Method

Three points are collinear if the value of the area of the triangle formed by the three points is zero.

Substitute the coordinates of the given three points in the area of the triangle formula. If the result for the area of the triangle is zero, then the given points are said to be collinear.

Formula for area of a triangle formed by three points is

\[\frac{1}{2}\begin{bmatrix}x_{1}-x_{2} & x_{2}-x_{3}\\ y_{1}-y_{2}& y_{2}-y_{3}\end{bmatrix}\]

Let us substitute the coordinates of the above three points R, S, and T in the determinant formula above for the area of a triangle to check if the answer is zero.

\[\frac{1}{2}\begin{bmatrix}2-4 & 4-6\\ 4-6& 6-8\end{bmatrix} = \frac{1}{2}\begin{bmatrix}-2 & -2\\ -2&-2 \end{bmatrix} = \frac{1}{2}(4-4)= 0\]

Since the result for the area of the triangle is zero, therefore R (2, 4), S (4, 6), and T (6, 8) are collinear points.

Non-Collinear Points Definition

The set of points which do not lie on the same straight line are said to be non-collinear points. In the below figure, points X, Y, and Z do not make a straight line, so they are called the non-collinear points in a plane.

Examples of Non-collinear Points

Let's consider P, Q, and R are non-collinear points, draw lines joining these points.

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Number of lines through these three non-collinear points is 3.

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If P, Q, R, & S are non-collinear points then

Number of lines through these four non-collinear points is 6.

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So in general, we can say number of lines through “n” non-collinear points = \[\frac{n(n-1)}{2}\]

Solved Examples

Example 1: Identify the collinear and non-collinear point from the below figure.

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Solution: Points A, B, and C are collinear. And points D, E, and F are non-collinear points in a plane.

Example 2: Check whether the points (2, 5) (24, 7), and (12, 4) are collinear or not?

Solution: Using slope formula to solve this problem.

Let the points be A (2, 5), B (24, 7), and C (12, 4).

Slope of AB = \[\frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{7-5}{10-2} = \frac{2}{8} = \frac{1}{4}\]

Slope of BC = \[\frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{4-7}{12-24} = \frac{-3}{-12} = \frac{1}{4}\]

As slope of AB = slope of BC. We can say that the given points A (2, 5), B (24, 7), and C (12, 4) are collinear.

Example 3: Given that the points P (8, 1), Q (15, 7), and R (x, 3) are collinear. Find x.

Solution: As given the points P, Q, and R are collinear, we have:

Slope of PQ = Slope of QR

\[\frac{7-1}{15-8} = \frac{3-7}{x-15}\]

\[\frac{6}{7} = \frac{-4}{x-15}\]

6 (x - 15) = - 4 * 7

6x - 90 = - 28

6x = 62

=> x = 31/3

Page 2

In rupees, one billion equals 10,000 lakhs. 1,000,000,000 is a natural number that equals one billion. The number 999,999,999 comes before 1 billion, and 1,000,000,001 comes after it. The concept of place value is used in mathematics to describe quantities. There are two ways to interpret the place value of the digits in a number. The Indian System and the International System are the two. The place value charts are used to determine the number's positional values. With the support of positions, numbers in the general form can be extended.

The place value is ordered from right to left. Starting with the unit location (one's place), the place value progresses to tens, hundreds, thousands, and so on. Let us look at the value of 1 billion in rupees of the Indian scheme of place value and 1 billion dollars in rupees in words. We'll also look at the position value chart for both the Indian and International systems.

Overview of the Topic

Students get scared by seeing big numbers but nothing to get scared of. It is important that students must understand how to identify big numbers. They must have proper knowledge and information about the rules for identifying big numbers. There are different ways to identify the numbers. Mainly, students are taught about two different systems of place value. One is the Indian system of place value and the other is the International System of place value. It is important that students must have knowledge of both the Indian and the International System of place values to identify big numbers easily. Students must know how to convert international numbers in the Indian system. The names given to different numbers in the Indian and international system are different and students must know this concept for excelling in higher classes.

1 Billion is a big number and it is equal to 10,000 lakhs in Indian money. One billion can be written as 1,000,000,000. Students can remember this by counting the zeroes. There are nine zeros in one billion. Similarly, if students want to remember other big numbers they can count the zeroes. For example, one million will have six zeros and it is written as 1000,000. Thus, in a similar manner students can remember other big numbers.

One Billion in Rupees

Consider 1 dollar = 73.80 rupees

Value of 1 billion = 1,000,000,000 rupees

So 1 billion dollars in rupees = 73.80 x 1,000,000,000 =7.38 x 101010

Similarly

5 billion dollars in rupees =73.80 x 5,000,000,000 = 369 x 101111

1 Billion in Indian Rupees

The International System uses billions based on position value charts. The equivalent value of 1 billion in Indian rupees (according to the Indian System) is

1 billion in rupees = 1,000,000,000 Rupees

We can write it as:

1 Billion = 10,000 Lakhs (As we know 1 lakh = 1,00,000)

As a result, a billion in lakhs equals 10,000 lakhs. It means that one billion lakhs equals ten million lakhs.

In other words, 1 billion equals 100 crores (as 1 lakh equals 1,00,00,000).

Conversion from Billion to Lakhs

Multiply the given billion value by ten thousand lakhs to convert the given billion value to lakhs (10,000 Lakhs)

Multiply 7 by 10,000 lakhs, for example, to convert 7 billion to lakhs.

(i.e.,) 7 Billion = 7 x 10,000 lakhs

70,000 lakhs = 7 billion

As a result, 7 billion equals 70,000 lakhs.

1 billion dollar to inr is 73,80,00,50,000

Conversion from Billion to Crores

Multiply the given billion value by 100 crores to convert the given billion value to crores.

To convert 9 billion to crores, multiply 9 by 100 crores, as an example.

(To put it another way,) 9 billion is equal to 9 x 100 crores.

900 crores = 9 billion dollars

As a result, 9 billion equals 900 crores.

Similarly, any billion value can be converted to values in the Indian numbering system, such as lakhs, crores, and so on.

How Many Zeros in a Billion?

There are nine (9) zeros in a billion.

1 billion = 1,000,000,000

How Many Millions is a Billion?

1 million = 1000,000

One million is equal to 1000 thousand.

1 billion = 1000 million = 1000,000,000

Therefore, one billion is equal to 1000 million.

Place Value Chart for Indian System

The sequence of the position value of the digit in the Indian system is as follows:

Crores

Lakhs

Thousands

Ones

Ten Crores

Crores

Ten Lakhs

Lakhs

Ten Thousands

Thousands

Hundreds

Tens

Ones

10,00,00,000

1,00,00,000

10,00,000

1,00,000

10,000

1000

100

The Hindu-Arabic method of numeration is also known as the Indian system of numeration. The comma symbol "," is used to differentiate the intervals in this scheme. The first comma appears after three digits from the right hand, followed by two digits, two digits, and then every two digits.

International System Place Value Chart:

The sequence of a digit's position value in the International System is as follows:

Ones

Thousands

Thousands	Ten Thousand	Hundred Thousand
1000	10,000	100,000

Millions

One Million	Ten Million	Hundred Million
1000,000	10,000,000	100,000,000

Billions

One Billion

Ten Billion

Hundred Billion

10,000,000,000

100,000,000,000

How to Use the Calculator to Convert Billion to Rupees?

The following is the protocol for using the billion to rupees conversion calculator:

Step 1: In the input region, type the number of billions.

Step 2: To get the conversion value, press the "Convert" button.

Step 3: Finally, in the output sector, the value of the conversion from billions to rupees will be shown.

What Does Billion to Rupees Conversion Mean?

In the Indian and International (more precisely the US) numeral systems, the place value of digits is referred to in various ways. Digits in the Indian system have place values of Ones, Tens, Hundreds, Thousands, Ten Thousand, Lakhs, Ten Lakhs, Crores, and so on. The position values of digits in the International system are in the order Ones, Tens, Hundreds, Thousands, Ten Thousand, Hundred Thousands, Millions, Billions, and so on. As a result, 1 billion is converted to 100 crores in the conversion from billions to rupees.

1 billion rupees = 1,000,000,000 rupees

Since 1 lakh equals Rs. 100000, 1 billion equals 10,000 lakhs.

100 Crores = 1 Billion

Solved Examples

1. What is the Rupee Equivalent of 5 Billion?

Solution: We know that a billion rupees equal 1,000,000,000 rupees.

As a result, the rupee value of 5 billion is estimated as follows:

5 Billion = 5 x 1,000,000,000

5 Billion = 5,000,000,000 rupees

We may also assume that 5 billion equals 500 crores.

2. In Crores, What is the Worth of 4.6 Billion?

Solution: We know that

1 Billion = 100 crores

Therefore, 4.6 Billion = 4.6 x 100 crores

4.6 Billion = 460 crores

Hence, the value of 4.6 billion is 460 crores.

Overview of Place Value

Place value is an important concept in mathematics. It helps to determine the position of a digit in the given number. Each digit in a number has a position. A number can be expanded depending on the position of different digits. We count the place value of digits from right to left. The position will start from the unit's place and move on to tens, hundreds, thousands, ten thousand, etc.

The place value of every digit in a given number is different. A number may have two same digits but both digits in the number will have a different position or place value. For example, 5456 in this number 5 will have a different place value. The number on the right will have a place value of tens and the number on the left will have a place value of ten thousand.