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Concurrent Lines occur when three or more lines intersect at a single point. This concept is a significant topic within the realm of straight lines. In this article on Concurrent Lines, we will delve into the precise definition of concurrent lines, explore the conditions that lead to concurrent lines, examine the role of concurrent lines within triangles, distinguish concurrent lines from intersecting lines, and solve some practice problems to get a better understanding of the topic.
Table of Content Meaning of Concurrent LinesConcurrent lines are a set of three or more straight lines that intersect at a common point. The point at which they intersect is called the point of concurrency. Concurrent lines are always going to be non-parallel and the number of lines passing through the point of concurrency must be equal to or greater than three. If the number of lines intersecting at a common point is only two then it is considered as an intersecting line. Concurrent Lines Definition
Point of ConcurrencyPoint of Concurrency is a term used to describe a point where three or more lines, rays or line segments intersect. In simple words, a point of intersection of concurrent lines is known as a Point of Concurrency. .png) Concurrent Lines In various geometric shapes, different types of points of concurrency can be identified. Some well-known points of concurrency for triangles are:
We will these points further in the article. Examples of Concurrent LinesWe know that any three lines with a common intersection point are called concurrent lines. These concurrent lines are also present in various geometrical shapes. Let’s see examples of such concurrent lines.
Triangle: Line segments that pass through the vertex and meet at a common point is one of the examples of concurrent lines in a triangle. .png) Example: Point of Concurrency in Triangle Condition for Three Lines to be ConcurrentThere are two conditions for three lines to be concurrent. First is if the determinant of the coefficients of the equations of the line is zero then line is concurrent and second is solving two equations of lines and putting the value of x and y in the third equation and checking if LHS = RHS. Let’s learn these methods in detail. Condition 1: Determinant MethodIf three lines are considered concurrent, it means that the point where two of these lines intersect also lies on the line. Assume the equation of three lines as: a1x + b1y + c1 = 0 ………….(i) a2x + b2y + c2 = 0 ………….(ii) a3x + b3y + c3 = 0 ………….(iii) The criterion that determines whether three lines are concurrent can be expressed as follows: [Tex]\begin{vmatrix} a_{1} & b _{1}& c_{1}\\ a_{2}& b_{2} & c_{2}\\ a_{3}& b_{3} & c_{3} \end{vmatrix} = 0 [/Tex] OR a1(b2c3 – c2b3) – b1(a2c3 – a3c2) + c1(a2b3 – a3b2) = 0 Condition 2: Substitution MethodIf three lines are considered concurrent, it means that the point where two of these lines intersect also lies on the line. We can follow these steps to prove it. Assume the equation of three lines as: a1x + b1y + c1 = 0 ………….(i) a2x + b2y + c2 = 0 ………….(ii) a3x + b3y + c3 = 0 ………….(iii) Step 1: Solve equations (i) and (ii) using the substitution method to get the values of x and y. Step 2: Now put the values of x and y in equation (iii) if the final value becomes zero then the lines are concurrent else they are not concurrent. Concurrent Lines, Line Segments and RaysLine segments are lines that have a starting and an ending point. When a set of three or more line segments meet at a point is called a concurrent line segment. Rays are a type of line that has only a single point and no endpoint. When a set of three or more rays meet at a point it is called a concurrent ray.
Concurrent Lines vs Intersecting LinesIntersecting lines are formed when only two lines are meeting at a common point. The point at which both lines meet is commonly known as Intersecting Point. It is important to note that this line can also be replaced with line segments or rays.
Concurrent Lines in a TriangleA triangle is a 2D geometric shape that has three sides and angles. In a triangle, there are four most common sets of concurrent lines.
Also, Read Solved Examples on Concurrent LinesExample 1: Prove the following set of three lines are concurrent. 15x – 18y + 1 = 0, 12x + 10y -3 = 0 and 6x + 66y – 11 = 0.Solution:
Example 2: Find the value of c for which the three lines are concurrent 2x – 5y + 3 = 0, 5x – 9y + c = 0, x – 2y + 1 = 0 .Solution:
Example 3: Prove the following set of three lines are concurrent M1 = (a-b)x + (b-c)y + (c-a) = 0, M2 = (b – c)x + (c-a)y + (a-b) = 0, M3 = (c-a)x + (a-b)y + (b-c) = 0. Solution:
Example 4: The following set of three lines are concurrent a1x + b1y + 1 = 0, a2x + b2y + 1 = 0, and a3x + b3y + 1 = 0. Prove that the points (a1 , b1), (a2 , b2), (a3 , b3) are collinear. Solution:
Practice Problems on Concurrent LinesQ1. Prove the following set of three lines are concurrent 3x – 5y – 11 = 0, 5x + 3y -7 = 0 and x + 2y = 0. Q2. Prove the following set of three lines are concurrent x = 0, y = 1, x = y. Q3. Find the conditions at which these three lines meet at a common point y1 = m1x + c1, y2 = m2x + c2 and y3 = m3x +c3. Q4. Find the conditions at which these three lines meet at a common point M1 = (a+b)x + cy + 1, M2 = (b+c)x + ay +1, M3 = (c+a)x + by + 1 FAQs on Concurrent Lines1. What are Concurrent lines?
2. What is the Point of Concurrency?
3. Why Parallels can’t be Concurrent?
4. What is the difference between Concurrent Lines and Intersecting Lines?
5. What are some Real-Life Applications of Concurrent Lines?
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Reffered: https://www.geeksforgeeks.org
| Class 6 |
Type: | Geek |
Category: | Coding |
Sub Category: | Tutorial |
Uploaded by: | Admin |
Views: | 14 |