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 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
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.
3. Show that the straight lines 2a - 3b +4 = 0, 9a + 5b = 19 and 2a -7b + 12 = 0 are concurrent. Find the point of concurrency.
The three given equations are-
2a - 3b + 4 = 0 ------ i
9a + 5b - 19 = 0 ------- ii
2a - 7b + 12 = 0 ----- iii
Solving the first equation, we get-
a= -4 + 3b/2
Putting the value of a in the second equation we get-
9 (-4 + 3b/2 ) + 5b - 19= 0
-36/2 + 27b/2 + 5b = 19
-18 + 27b + 10b/2 = 19
37b/2 = 37
b= 2
Therefore, a= 1
So, the point of intersection of the first two lines is (1, 2)
Now, we have to apply this to the third line. By doing so we get,
2(1) - 7(2) + 12= 0
2 - 14 + 12 = 0
-12 + 12 = 0
0 = 0
Therefore, the given lines are concurrent and their point of intersection is (1,2).
4. If the lines 2p+q–3=0, 5p+kq–3=0 and 3p–q–2=0 are concurrent, find the value of k.
We know that concurrent lines are those lines which pass through the same point. Since, the given lines are concurrent, the point of intersection of the points of concurrency of the lines will be the same.
2p + q - 3 = 0 --------i
5p + kq - 3= 0 ------ ii
3p - q - 2= 0 -------iii
By solving the first equation, we get the value of q,
q= 3- 2p
Now, putting this value of q in the third equation, we get
3p - (3- 2p) - 2 = 0
3p -3 + 2p - 2 = 0
5p = 5
p= 1
Therefore, q = 1
The intersection point is (1,1)
Now this intersection point will coincide with the second equation as the lines are given to be concurrent. Therefore, by putting the point (1,1) in the second equation, we get
5(1) + k(1) = 3
5 + k = 3
k= -2
Therefore, the value of k is -2.