Coincident lines

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The coincident lines are those that share all their points in common, that is, they have the same inclination and go through the same coordinates in the Cartesian plane.

The coincident lines, from the graphic point of view, are drawn one on top of the other, both being identical.

Likewise, it should be mentioned that no angles are formed between coincident lines, as is the case with perpendicular lines, which form four 90º angles, and oblique lines, which form two acute angles (less than 90º) and two angles. obtuse (over 90º).

Another important point is that the parallel lines, like the coincident ones, comply with having the same inclination (slope), but they do not have any point in common.

We must also specify that a line is a one-dimensional geometric element that consists of an infinite series of points that go in a single direction, that is, it does not present curves.

How to know if two lines are coincident?

To explain how to determine if two or more lines are coincident, we must first remember that, from analytic geometry, a line can be expressed as a first-order equation like the following:

y = mx + b

Thus, in the equation y is the coordinate on the ordinate axis (vertical), x is the coordinate on the abscissa axis (horizontal), m is the slope (inclination) that forms the line with respect to the abscissa axis , and b is the point where the line intersects the ordinate axis.

The above is the explicit equation of a line. If two or more lines have the same explicit equation, they are coincident.

However, we can also do a broader analysis, with the implicit equations of two lines that would have the following form:

0 = Ay + Bx + C

As we can see, it is an equation similar to the one in the lines above, but next to the equality we leave 0.

So, A is the coefficient that will be multiplied by the coordinate on the vertical axis, B is the coefficient that will be multiplied by the coordinate on the horizontal axis, and C is multiplied by 1.

Having all this information, two (or more) lines are coincident when their coefficients are proportional, that is, limiting ourselves to the case of two lines we would have:

A / A ’= B / B’ = C / C ’

In the above equation A, B and C are the coefficients of a line, while A ', B' and C 'are the coefficients of their coincident line.

Example of coincident lines

Suppose we have two lines with the following implicit equations:

Line 1: 0 = 9y-3x + 8

Line 2: 0 = 27y-9x + 24

So we divide the coefficients:

9/27=1/3

3/9=1/3

8/24=1/3

Therefore, line 1 and line 2 are coincident.

In the image below, we see two other lines that coincide with their respective equations:

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