Concurrent Lines
Three or more lines in a plane passing through the same point are concurrent lines. Whenever two nonparallel lines meet each other they form a point of intersection. When a third line also passes through the point of intersection made by the first two lines then these three lines are said to be concurrent lines. The point of intersection of all these lines is called the 'Point of Concurrency'. For example, we can see that three altitudes that are drawn on a triangle intersect at a point, which is called 'Orthocenter'. It is to be noted that only nonparallel lines have a point of concurrence since they extend indefinitely and meet at a point.
| 1. | Concurrent Lines Definition |
| 2. | Concurrent Lines of a Triangle |
| 3. | Solved Examples on Concurrent Lines |
| 4. | Practice Questions on Concurrent Lines |
| 5. | FAQs on Concurrent Lines |
Concurrent Lines Definition
Concurrent lines are defined as the set of lines that intersect at a common point. Three or more lines need to intersect at a point to qualify as concurrent lines. Only lines can be concurrent, rays and line segments can not be concurrent since they do not necessarily meet at a point all the time. There can be more than two lines that pass through a point. A few examples are the diameters of a circle are concurrent at the center of the circle. In quadrilaterals, the line segments joining midpoints of opposite sides, and the diagonals are concurrent.
How to Find If Lines are Concurrent?
To find if three lines are concurrent or not, there are two methods. Let us discuss both of them.
Method 1:
Let us consider three lines,
Line 1 = \(a_{1}x\) + \(b_{1}y\) + \(c_{1}z\) = 0 and
Line 2 = \(a_{2}x\) + \(b_{2}y\) + \(c_{2}z\) = 0 and
Line 3 = \(a_{3}x\) + \(b_{3}y\) + \(c_{3}z\) = 0.
To conclude if the above three lines are concurrent, the following condition shown below as a determinant should be evaluated to 0.
One other method to check if the lines intersect each other is as follows.
Method 2:
To check if three lines are concurrent, we first find the point of intersection of two lines and then check to see if the third line passes through the intersection point. This will ensure that all three lines are concurrent. Let us understand this better with an example. The equations of any three lines are as follows.
4x - 2y - 4 = 0 ----- (1)
y = x + 2 ----- (2)
2x + 3y = 26 ----- (3)
Step 1: To find the point of intersection of line 1 and line 2, solve the equations (1) and (2) by substitution method.
Substituting the value of 'y' from equation (2) in equation (1) we get,
4x - 2 (x + 2) - 4 = 0
4x - 2x - 4 - 4 = 0
2x - 8 = 0
x = 8/2
x = 4.
Substituting the value of 'x = 4' in equation (2), we get the value of 'y'.
y = x + 2 ----- (2)
y = 4 + 2
y = 6
Therefore, line 1 and line 2 intersect at a point (4,6).
Step 2: Substitute the point of intersection of the first two lines in the equation of the third line.
Equation of the third line is 2x + 3y = 26 ----- (3)
Substituting the values of (4,6) in equation (3), we get,
2(4) + 3(6) = 26
8 + 18 = 26
26 = 26
Therefore, the point of intersection goes right with the third line equation, which means the three lines intersect each other and are concurrent lines.
Concurrent Lines of a Triangle
A triangle is a two-dimensional shape that has 3 sides and 3 angles. Concurrent lines can be seen inside triangles when some special type of line segments are drawn inside them. Be it any type of triangle, we can locate four different points of concurrence. They are,
Incenter: The point of intersection of three angular bisectors inside a triangle is called the incenter of a triangle.
Circumcenter: The point of intersection of three perpendicular bisectors inside a triangle is called the circumcenter of a triangle.
Centroid: The point of intersection of three medians of a triangle is called the centroid of a triangle
Orthocenter: The point of intersection of three altitudes of a triangle is called the orthocenter of a triangle.
Topics Related to Concurrent Lines
Check out some interesting topics related to concurrent lines.
Solved Examples on Concurrent Lines
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Example 1: Verify whether the following lines are concurrent or not. The line equations are, x + 2y - 4 = 0, x- y - 1 = 0 , 4x + 5y - 13 = 0.
Solution:
To check if three lines are concurrent, the following condition should be satisfied.
Comparing the given three line equations to \(a_{1}x\) + \(b_{1}y\) + \(c_{1}z\) = 0, \(a_{2}x\) + \(b_{2}y\) + \(c_{2}z\) = 0 and \(a_{3}x\) + \(b_{3}y\) + \(c_{3}z\) = 0, let us find the values of \(a_{1}\), \(b_{1}\), \(c_{1}\) , \(a_{2}\), \(b_{2}\), \(c_{2}\) , \(a_{3}\), \(b_{3}\), \(c_{3}\)
\(a_{1}\)= 1 \(b_{1}\)= 2 \(c_{1}\)= -4
\(a_{2}\)= 1 \(b_{2}\) = -1 \(c_{2}\) = -1
\(a_{3}\) = 4 \(b_{3}\) = 5 \(c_{3}\) = -13
Arranging them in the determinants form, we get,
On solving this, we get,
= 1(13 + 5) - 2(-13 + 4) - 4(5 + 4)
= 18 + 18 - 36
= 36 - 36
= 0The above condition holds good for the three lines. Therefore, the three lines are concurrent.
FAQs on Concurrent Lines
What are Concurrent Lines?
Concurrent lines are the lines that have a common point of intersection. Only lines intersect each other to form concurrent lines as they extend indefinitely and therefore meet at a point.
Are Parallel Lines Concurrent?
No parallel lines can not be concurrent lines, because they never meet at any point. Even when parallel lines are extended indefinitely they can not be concurrent lines, since they will not have a common point at which they intersect.
How Do You Know If a Line is Concurrent?
There should be at least three lines to define a set of concurrent lines. If two lines intersect they meet at a point. If a third line also passes through this intersection point then we can say that the three lines are concurrent.
What is the Difference Between Intersecting Lines and Concurrent Lines?
Three lines meet at a point to form concurrent lines. The meeting point is called the 'point of concurrence'. When two lines meet at a point, they are called intersecting lines. The meeting point of these two lines is called the 'point of intersection'.
If More Than 3 Lines Intersect at a Point Can They be Called Concurrent Lines?
Yes, more than three lines intersecting at a point can also be called concurrent lines since they all share a common point of intersection.
What are the Concurrent Lines of a Triangle?
There are 4 concurrent lines for a triangle. They are Incenter, circumcenter, centroid, and orthocenter. They are the points of intersection formed when the 3 angle bisectors, 3 perpendicular bisectors, 3 medians, and 3 altitudes of a triangle concur at a point respectively.