[Maths Class Notes] on Concurrent Lines Pdf for Exam

It is commonly known that two non-parallel intersect at one point. If a third line is formed passing through one common point or intersecting each other at one common point, then these straight lines are termed as concurrent lines. The word ‘concurrent ‘ means something that occurs at the same time or same point.

By Eculid’s Lemma, it is stated that two lines have a maximum one common point of intersection. In the figure given below, we can see that lines are meeting each other at point P. When three or more lines intersect together exactly at one single point in a plane then they are termed as concurrent lines. The point where three or more lines meet each other is termed as the point of concurrency. 

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In this article, we will discuss concurrent lines, concurrent lines definition, concurrent line segments and rays, differences between concurrent lines and intersecting lines etc.

Concurrent Line Definition

A set of three or more lines are termed as concurrent when passing through one common point or coincide exactly at one common point. The common point where all the lines intersect or coincide is known as the point of concurrency. In the figure given below, the line shown in blue, orange, and black is passing through the point O. Hence, all three lines are concurrent to each other.

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Concurrent Line Segments and Rays

When three or more line segments are intersecting each other at one single point then these lines are determined as concurrent line segments. In the figure given below, AB ,CD and EF are three line segments intersecting each other at point O. Hence, it can be said that concurrency can also be applied to line segments. 

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When three or more rays in a two – dimensional plane intersect each other at one single point, then they are termed as concurrent rays. The common point where all the rays meet each other is termed as the point of concurrency for all the rays. In the figure given below, three rays PQ, RS and MN which are meeting each other at point O are concurrent with each other.

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Point of Concurrency

When three or more lines pass through a same point they are called concurrent lines. The point where they all meet or intersect is called the point of concurrency. For example if line A, line B and line C are concurrent lines which meet at a point say Z, then Z would be the point of concurrency where the three lines meet each other. 

Concurrent Lines of a Triangle

A triangle is a 2D shape with three sides and three angles. There can be concurrent lines in a triangle if line segments are drawn inside a triangle. A triangle has four different concurrency points irrespective of the type of the triangle. These four points are- 

1. Incenter- This is a point of intersection of the three angular bisectors (lines dividing the angles into two equal parts) inside a given triangle. 

2. Circumcenter- This is a point of intersection of the three perpendicular bisectors ( lines that divide a given line into two equal parts at right angle) and inside a triangle. 

3. Centroid- This is a point of intersection of the three medians (line joining the vertex to the midpoint of the opposite side) of a given triangle. 

4. Orthocentre- This is the point of intersection of the three altitudes (line joining the vertex to the  opposite side and is perpendicular) of a triangle. 

Difference Between Concurrent Lines and Intersecting Lines

As we know that if three or more lines, line segments, or rays meet each other at one common point then they are said to be in concurrency. But in the case of intersecting lines, there are only two lines, line segments or rays that meet each other at one common point.

Below are some points which show differences between concurrent lines and intersecting lines in tabulated form

Quiz Time

1. What is Represented as the Point of Concurrency for the Median of a Triangle?

1. Circumcentre

2. Centroid

3. Incener

4. Orthocenter

     2.  Three or More Lines Are Considered as Concurrent I That Pass Through

1. Same line

2. Same point

3. Same plane

4. None of the above
   

Solved Examples

1. Show That the Three Lines 2p – 4q + 5 = 0, 7p – 8q + 5 and 4p + 5q = 45 Are Concurrent Lines and Also Determine the Point of Concurrency.

Solution:

Let,

2p – 4q + 5 = 0   ……… (1)

7p – 8q + 5 = 0 ……….(2)

4p + 5q = 45 …………(3)

Let us use the substitution method and solve equations 1 and 2 given above.

1. 2p + 5 =  4q    

2. 7p – 8q + 5 = 0  or 7p – 2(4q) + 5 = 0 

Now substitute 4q = 3p + 5……..(3)

7p – 2(3p + 5) + 5 = 0

7p – 6p – 10 + 5 = 0

p – 5 =0

p = 5

Hence, the point of intersection of lines 1 and 2 is (5,5). 

Now , let us examine whether the third line satisfies the point (5,5).

i.e. 4p + 5q = 45

(3) → 4(5) + 5(5) = 45

20 + 25 = 45

45 = 45

Hence, all three given lines are passing through the point (5,5) and they are said to be concurrent lines. And the point of concurrency is (5,5).

2. Verify, If the Following Lines are Concurrent

p₁x + q₁y + r₁ = 0…………….(1)

 p₂ x + q₂ y + r₂ = 0………(2)

( 2p₁ – 3p₂)x + ( 2q₁ – 3q₂)y + ( 2r₁ – 3r₂) = 0……(3).

Solution:

If we carefully see the above three lines, we will notice that if the given lines are represented by  L₁ , L₂ and L₃, then we have L₃ – 2L₁ + 3L₂ . Hence, we have three constants, not all zero such that pL₁ + qL₂ + rL₃ = 0. Therefore, the given lines are concurrent.