Coincident Lines

Lines were invented by ancient mathematicians to represent straight things with minimal width and depth. Euclid defined lines as a length less than a breadth. Lines are the foundation of Euclidean geometry. An angle is formed when two rays (parts of a straight line) intersect in the same plane.

The shapes we see all around us are made up of an unlimited number of points. A straight line is formed when a point moves in such a way that its direction remains unchanged. In other terms, a line is represented by a one-dimensional collection of dots stretching infinitely in both directions, as seen below. A line never comes to an end.

Definition:-

Lines that cross or lay on top of each other are referred to as coincident lines. In Geometry, you may have learnt about several sorts of lines, such as parallel and perpendicular lines, and how they relate to a two-dimensional or three-dimensional plane.

Parallel lines are those that are perpendicular to one other and have a set distance between them. Perpendicular lines, on the other hand, are lines that meet at 90 degrees. Parallel and perpendicular lines, on the other hand, do not intersect.

Aside from these three lines, there are others that are not parallel, perpendicular, nor coincide. They could be oblique lines or intersecting lines that cross at different angles rather than perpendicularly.

The term ‘coincide’ refers to when something happens at the same time. In mathematics, coincident lines are lines that lay on top of each other in such a way that they appear to be a single line rather than double or many lines when seen. Although the figure of coinciding lines looks to be a single line, we actually have two lines there.

• Coincident lines equation:-

When considering a line’s equation, the conventional form is:

                    y = mx + d

The slope of the line is represented as m, and the intercept is dd

• Parallel Line Equation:

Now, in the instance of two parallel lines, the equations of the lines are written as:

y = m1x + d1

y = m1x + d2

Parallel lines are, for example, y = 4x + 2 and y = 4x + 5. Both lines have a slope of 4 and an intercept difference of 3. As a result, at a distance of three units, they are parallel.

• Coincident Lines Equation:

The equation for lines is provided by; when we talk about coincident lines, the equation for lines is given by;

ax + by = c

There must be no difference in intercept between two lines that coincide.

For example, x + 2y = 3 and 2x + 4y = 6 are the two lines. 

The second line is twice as long as the first.

Because we get; if we put ‘y’ on the left-hand side and take the rest of the equation on the right-hand side, then we get;

Line 1:  2y = 3 – x ———– (1)

Line 2: 4y = 6 – 2x  

2.2y = 2 (3 – x) 

2y = 3 – x ———- (2)

Comparing both equation (1) and (2),

We can see that both lines are identical.

As a result, they are coincident lines.

Coincidence lines formula:-

Assume that A1x + B1y + C1 = 0 and A2x + B2y + C2 = 0 are pairs of linear equations with two variables. If the lines that represent these equations are parallel, they are said to be coincident, then

A1/A2=B1/B2=C1/C2

The above pair of equations is said to be consistent, and it can have an endless number of solutions i.e., the two lines are coincident.

Difference between parallel and coincident lines:-

Parallel lines have the same width or have a constant gap between them and never collide, however coincident lines do not have a constant space and are on top of each other, or in other words, one line entirely covers the other line. There are no shared points on parallel lines, but there are an infinite number of shared points on coincident lines. The convergence of a line and a line might be the unoccupied set, a point, or a line in Euclidean mathematics. Recognizing these situations and watching the convergence point has been employed in a variety of applications, including computer graphics, movement planning, and impact placement.

When two lines are not in the same plane, they are termed slant lines, and they have no point of convergence in three-layered Euclidean math. If they are on the same plane, there are three possibilities: if they match (are not distinct lines), they share an infinite number of focuses for all intents and purposes (to be specific, each of the focuses on both of them); if they are distinct but have a similar incline, they are supposed to be equal and share no focuses practically; in any case, they have a single mark of the crossing point

The number and areas of potential crossing sites between two lines, as well as the quantity of prospective lines with no convergences (equal lines) with a given line, are distinguishing features of non-Euclidean mathematics.

Conclusion:-

Lines that coincide or be on top of one another are known as coincident lines. We’ve taught about several sorts of lines in Geometry so far, such as parallel and perpendicular lines, and how they relate to a two-dimensional or three-dimensional plane. Parallel lines are lines that are parallel to each other and are separated by a defined distance.