How to Identify Concurrent Lines with Step-by-Step Methods
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-
Incenter- This is a point of intersection of the three angular bisectors (lines dividing the angles into two equal parts) inside a given triangle.
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.
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.
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
What is Represented as the Point of Concurrency for the Median of a Triangle?
Circumcentre
Centroid
Incener
Orthocenter
2. Three or More Lines Are Considered as Concurrent I That Pass Through
Same line
Same point
Same plane
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.
2p + 5 = 4q
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.
FAQs on Concurrent Lines in Mathematics
1. What are concurrent lines in mathematics?
In geometry, when three or more lines in a plane pass through a single, common point, they are known as concurrent lines. This shared point is a fundamental property that distinguishes a set of concurrent lines from lines that intersect at different points. For example, while any two non-parallel lines will intersect, it is a special condition for a third line to pass through that exact same intersection point.
2. What is the point of concurrency and can you give an example?
The point of concurrency is the specific point where three or more concurrent lines intersect. A classic example from geometry involves a triangle: the three medians of any triangle are concurrent, and their point of concurrency is called the centroid, which is also the triangle's centre of mass.
3. How do you determine if three lines are concurrent?
There are two primary methods to check for concurrency of three lines given by equations L₁, L₂, and L₃:
- Method 1: Solving Systematically
Find the point of intersection (x, y) by solving the equations for any two of the lines (e.g., L₁ and L₂). Then, substitute this (x, y) point into the equation of the third line (L₃). If the point satisfies the third equation, the lines are concurrent. - Method 2: Using Determinants
For three lines in the form a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0, and a₃x + b₃y + c₃ = 0, they are concurrent if and only if the determinant of their coefficients is zero. That is, |a₁ b₁ c₁; a₂ b₂ c₂; a₃ b₃ c₃| = 0.
4. What is the difference between concurrent, parallel, and intersecting lines?
These terms describe the relationships between lines in a plane:
- Concurrent Lines: Involves three or more lines that all pass through a single common point.
- Intersecting Lines: Typically refers to two lines that cross each other at exactly one point.
- Parallel Lines: These are lines in a plane that never meet, no matter how far they are extended. They have the same slope and zero points of intersection.
5. Are any three non-parallel lines in a plane always concurrent?
No, this is a common misconception. While any two non-parallel lines must intersect at one point, a third line is not guaranteed to pass through that same point. For three lines to be concurrent, they must satisfy a specific geometric or algebraic condition. If they don't, the three lines will typically intersect at three different points, forming a triangle.
6. What are some important examples of concurrent lines found in a triangle?
Triangles provide several key examples of concurrency, leading to important triangle centres:
- The three altitudes (perpendiculars from a vertex to the opposite side) are concurrent at the orthocenter.
- The three medians (lines from a vertex to the midpoint of the opposite side) are concurrent at the centroid.
- The three perpendicular bisectors of the sides are concurrent at the circumcenter.
- The three angle bisectors are concurrent at the incenter.
7. Can you provide a real-world example of concurrent lines?
Yes, a great real-world example of concurrent lines is the spokes of a bicycle wheel. All the spokes radiate from the central hub, meeting at a single point of concurrency. Another example is the lines of longitude on a globe, which are all concurrent at the North and South Poles.
8. Why does a zero determinant of coefficients indicate that three lines are concurrent?
The determinant condition arises from linear algebra. A system of three linear equations with two variables (x and y) generally has no unique solution. The condition that the determinant of the coefficient matrix is zero signals that the system of equations is linearly dependent. Geometrically, this dependency means that one of the lines can be expressed as a combination of the other two, forcing all three lines to intersect at a single, common point.
