Concurrent Lines

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What are Concurrent Lines?

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 passes 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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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 case of intersecting lines, there are only two lines, line segments or rays meet each other at one common point.

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

Concurrent Lines

Intersecting Lines

Three or more lines in a plane meet each other at one common point are termed as concurrent lines.

Two lines in a plane intersect each other at one common point are termed as intersecting lines.

The common point where all the concurrent lines meet each other is termed as the point of concurrency.

The common point where all the intersecting lines meet each other is termed as the point of intersection.

Example

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Example

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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.

FAQ (Frequently Asked Questions)

1. What is the Meaning of the Intersection of Three Lines or Concurrency of Straight Lines?

Let us consider three straight lines whose equations are p₁x + q₁y + r₁ = 0, p2 x + q2 y + r2  and p3 x + q3 y + r3 = 0. These lines are considered as concurrent if the below -given conditions hold true.

Ih have three straight lines with L₁ = 0, L2 = 0 and L3 = 0, then these lines will be considered as concurrent line if there exists three constant p,q, and r not all zero such that  pL₁ + qL2 + rL3 = 0

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Generally, three lines are considered as concurrent lines if any one of the lines travels through the point of intersection of the other two lines.

2. Are Non-Parallel Lines Concurrent?

A set of curves or lines are determined as concurrent if they all meet at the same point. In the figure shown below, three lines are said to be congruent because they are meeting at the same point P.

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Hence, the point P is considered as “point of concurrency”. As lines can be extended indefinitely in both the directions unless they are parallel they will meet somewhere at the point. Hence, all non-parallel lines are concurrent. Ray and line segments may or may not be considered as concurrent even when not parallel.