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# Coincident|Definition & Meaning

**Definition**

Any **geometrical object** for example **two lines** that are said to be **exactly overlapping** or an **exact overlap** of each other is known as **coincident**. There is no **parallelism** or **perpendicularity** between them, but their **characteristics** are **perfectly similar**. In other words, coincident lines are an arrangement of geometric objects such that **one shape** completely **covers another, **making it **hard to distinguish** one from the other.

By **coinciding**, we mean that it **occurs simultaneously**. The mathematical **definition** of coincidence is **two lines** lying **adjacent** to **each other**. We see them as a **single line** **rather** than **multiple lines** when they are arranged in this way.

**Representation of Coincident Line**

Consider **A1, B1,** and **C1, D1** to be the two lines when we **place** the **two lines** exactly on **one another** so that it **looks** like an **overlap** of **two lines** then the line is said to be **coincident** as shown in the figure below.

As you can see from the above diagram, there have been **two** **parallel lines** drawn together, but they seem to be a single line when viewed from the top, but they are two distinct lines. **To get** a **third line** that is coincident, **we overlapped **the **two lines together**, **first** drawing a **blue line** and then drawing a **red line**.

**Properties of Coincident Lines**

- There are
**infinite****common points**between the two lines. - There is
**no constant space**between them. - The
**two lines cover each**other. - One line
**exactly overlaps**another line.

**Equation of Coincident Line**

The following equation can be used to** represent linear equations.**

**y = mx + c**

where** y** is the **intercept**, **m** is the **slope,** and **c** is the **constant.**

Consider the general equation

**Ax + By = C**

**Case 1: **Suppose that **A=2, B=2, C=4** then the above equation becomes

**2x + 2y = 4**

**Case 2: **Suppose that **A=8, B=8, C=16** then the above equation becomes

**8x + 8y = 16**

Solving Case 1 and Case 2 equations in that order:

**Case 1 Equation**

2x + 2y = 4

x + y = 2

**x = 2 – y**

**y = 2 – x**

**Case 2 Equation**

8x + 8y = 16

x + y = 2

**x = 2 – y**

**y = 2 – x**

**In** **conclusion**, we can clearly see that the **x and y values** of both **case 1** and **case 2** are **equal** so both of the equations are **coincident.**

**Equation of Coincident Line Graphically**

We have drawn two equation **2x + 2y =4** and **8x + 8y =16. **We can see that **when x=0:**

**Equation 1**becomes**2(0) + 2y =4**simplifying gives**y=2****Equation 2**becomes**8(0) + 8y = 16**simplifying gives**y = 2**

**when y=0:**

**Equation 1**becomes**2x + 2(0) = 4**simplifying gives**x = 2****Equation 2**becomes**8x + 8(0) = 16**simplifying gives**x = 2**

Which clearly shows that these **two lines** are **coincident lines**.

Figure 2 – Graphing coincident lines. The equation in red is four times the equation in blue, so they represent the same line.

**Difference Between Coincident Line and Parallel Line**

**Parallel lines** have the **same width,** whereas **coincident** lines always **intersect each other** because **parallel lines** have a **constant space** in between them, and they **never intersect**. A **coincidental line** is on top of another line and does **not** have **constant space** between them; in other words, **one line covers the other** completely. There are **no common points** between **parallel lines** and c**oincident lines** have an **infinite number of points** shared in common.

The **convergence** of two lines in **Euclidean geometry** may be a **point**, a **line**, or a vacant **set**. For instance, PC **illustrations**, movement **planning**, and impact location have been used to recognize these cases and observe the **convergence point**.

Euclidean math call **slant lines** **non-converging lines** if they **are not** in the **same plane**. **When** they are **on** a **similar plane**, there are **three possibilities**: assuming that **they are alike**, they share an **unlimited number of focuses**; in any case, they have a solitary mark of the **crossing point;** assuming they are particular and have an **identical incline,** they **should be equal**; but if they are particular, they have a **similar incline**.

Figure 3 – Parallel lines are not coincident!

Consider the above figure there are **no common points** between the two lines and there is **constant space** between them so these lines are referred to as **parallel lines**.

Figure 4 – Coincident lines. It looks like it is just one line, but these are two lines exactly on top of each other!

Consider the above figure there are **infinite common points** between the two lines moreover there is **no constant space** between them and they are **covering each other** so these lines are referred to as **coincident lines**.

**Solved Examples of Coincident Lines**

**Example 1**

Consider the two equations, find whether the equations are coincident or not:

**5x + 5y = 10** and **10x + 10y = 20**

**Solution**

**Equation 1: **5x + 5y = 10. Put **x=0: **

5(0) +5y =10

5y=10

**y=2**

**Put y=0: **

5x +5(0) =10

5x=10

**x=2**

**Equation 2: **10x + 10y = 20. Put **x=0: **

10(0) +10y =20

10y=20

**y=2**

**Put y=0: **

10x + 10(0) = 20

10x = 20

**x = 2**

The **x and y values** of both equations are the **same** so they are **coincidental lines**. **Graphically** if we plot the two equations we can see that **one equation** is exactly **overlapping** **another** equation and **sharing infinite points** between each other and also there is **no constant space** between them.

Figure 5 – Example one represents coincident lines, as presented in this graph.

**Example 2**

Consider the two equations to find whether the line is coincident or not.

**Solution**

**y = 3x + 3** and** y = 3x +6**

**Equation 1: **y = 3x + 3. Put **x=0:**

y=3(0) + 3

**y=3**

**Put y =0:**

0=3x+3

**x=-1**

**Equation 1: **y = 3x + 6. Put **x=0**

y=3(0) + 6

**y=6**

**Put y =0**

0=3x+6

**x=-2**

The **x and y values** of both equations are **different** so they are **not coincident** rather they are **parallel lines** with **slope 3** and **intercept 2.** We can see from the figure below both of the equations are parallel with **no common points** between them and **no intersection along** with a c**onstant space between** the two lines.

Figure 6 – Example two’s lines as they appear on graph. Notice the lines are parallel, meaning they are not coincident.

**Example 3**

Consider the two equations to find whether the line is coincident or not:

**x+y = 1** and** 2x + 2y = 2**

**Solution**

**Equation 1: **x + y = 1. Put** x=0:**

0+y=1

**y=1**

**Put y=0:**

x+0=1

**x=1**

**Equation 2: **2x + 2y = 2. Put** x=0:**

2(0)+2y=2

2y=2

**y=1**

**Put y=0:**

2x+2(0)=2

2x=2

**x=1**

The **x and y values** of both equations are the **same** so they are **coincidental lines**. Graphically if we plot the two equations we can see that one equation is **exactly overlapping** another equation and **sharing infinite points** between each other and also there is **no constant space between them**.

Figure 7 – Graphing the coincident lines in example three

*All mathematical drawings and images were created with GeoGebra.*