Concurrent lines are the lines, in 2-D geometry, which intersect each other exactly at one point. The meaning of concurrent is happening at the same time or point. The intersecting lines are always concurrent. Show
Concurrent lines are non-parallel lines and extend indefinitely at both the direction. They intersect each other at a point somewhere in the plane. Thus we can say all parallel lines are not concurrent lines. When two or more lines pass through a single point, in a plane, they are concurrent with each other and are called concurrent lines. A point that is common to all those lines is called the point of concurrency. This property of concurrency can also be seen in the case of triangles. All the intersecting lines or non-parallel lines are concurrent. But parallel lines are not concurrent at any point on a plane. In the figure given below, you can see the three lines are all crossing point O. Hence, all these three lines are concurrent with each other. Condition of Concurrent LinesIf three lines are concurrent, then the point of intersection of two lines lies on the third line. Suppose, the equations of three lines are: a1 x + b1y + c1 = 0 ……………. (1) a2 x + b2 y + c2 = 0 ……………. (2) a3 x + b3 y + c3 = 0 ……………. (3) Thus, the condition, if the three lines are concurrent to each other, is; \(\begin{array}{l}\left|\begin{array}{lll} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array}\right|=0\end{array} \) Solved ExampleQuestion: Find if the lines 2x – 3y + 5 = 0, 3x + 4y – 7 = 0 and 9x – 5y + 8 =0 are concurrent. Answer: Given the three lines are: 2x – 3y + 5 = 0 3x + 4y – 7 = 0 9x – 5y + 8 =0 Now, as per the condition of the concurrence of lines, we need to find the determinant of the coefficients. Hence, \(\begin{array}{l}\left|\begin{array}{lll} 2 & -3 & 5 \\ 3 & 4 & -7 \\ 9 & -5 & 8 \end{array}\right|=0\end{array} \) = 2(32 – 35) – (-3)(24 + 63) + 5(-15 – 36) = 2(-3) + 3(87) + 5(-51) = – 6 + 261 -255 = 0 Therefore, the given three lines are concurrent. Concurrent Lines in TrianglesIn a triangle, the concurrent lines are:
Concurrent Line Segments and RaysWhen three or more line segments, intersect each other at a single point, then they are said to be concurrent line segments. See the figure below, where AB, CD and EF are three line segments and are intersecting each other at one point O. Hence, we can apply the concurrency to line segments also. When three or more Rays in 2-D plane cuts or meets at a single point, then they are called Concurrent Rays. The single point is the point of concurrency for all the rays. In the below figure, three rays PQ, RS and MN, which are intersecting at a point O, are concurrent to each other. Difference Between Concurrent Lines and Intersecting LinesAs we have already understood, if any three lines or line segments or rays are having a single intersection point, they are said to be in concurrency. While, in the case of intersecting lines, there are only two lines or line segments or rays that intersect with each other. We can write the differences in a tabular form.
Stay tuned with BYJU’S – The Learning App and also download the app for more interesting Maths-related concepts and personalized videos. Related ArticlesFrequently Asked Questions on Concurrent LinesWhen two or more lines intersect at a common point in a plane, then they are called concurrent. No, parallel lines are
not concurrent lines, because they do not intersect each other. The point of intersection of two or more lines that is common to all the lines is called point of concurrency. Yes any two intersecting lines are always concurrent. Medians of triangle intersect each other at a common point. Thus, they are also referred to as concurrent and the common point where they intersect is the centroid of the triangle. What is a line with one point called?A segment is named by its two endpoints, for example, ¯AB . A ray is a part of a line that has one endpoint and goes on infinitely in only one direction. You cannot measure the length of a ray.
What is a line that meets?Lines which meet or appear to meet when extended are called intersecting lines and the point where they meet is called the point of intersection.
What do you call the two lines that do not meet at one point?Two lines which do not meet at a point are called parallel lines.
What is Interseting line?Two or more lines which share exactly one common point are called intersecting lines. This common point exists on all these lines and is called the point of intersection. It is to be noted that: The intersecting lines meet at one, and only one point, no matter at what angle they meet.
|