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Elements of the circumference

The circumference is a locus in which all points are equidistant from a fixed point called the center.

Figure 1. Parts of the circumference

In a mathematical way the circumference is represented by an equation of second degree:

x^{2} + y^{2} = r^{2}

Where the previous equation represents a circle whose center is the origin of the coordinate axis system.

And to represent a circumference with the center outside the origin, we use the following equation:

\left(x - (h) \right)^{2} + \left(y - (k) \right)^{2} = r^{2}

Where h and k represent the center of the circumference at \left( h, k \right).

Let’s analyze two examples of the circumference:

Circumference example 1. Draw the locus represented by the equation x^{2} + y^{2} = 9.

As we can see in the equation of the exercise, it is a circle whose center is the origin and the radius we have to calculate, in this case:

r^{2} = 9

We just have to apply square root to the above equality to get our radius:

r = 3

Finally we can locate our circumference in the Cartesian plane:


Figure 2. Circumference x^{2} + y^{2} = 9

Let’s see an example where the center isn’t at the origin.

Example of a circumference 2. Draw the locus represented by the equation (x + 1)^{2} + (y - 1)^{2} = 5.

What we can observe in this exercise is that it is not a circumference that is at the origin, if we don’t have much practice, we have to place the appropriate parentheses to the equation of the circumference as shown below:

(x + 1)^{2} + (y - 1)^{2} = (x - (-1))^{2} + (y - (1))^{2} = 5

Perfect, now that we have added the proper parentheses, we can say that the center is at the point:

(-1 , 1)

It would only be necessary to calculate the radius of the circumference, for that we simply have to apply the square root to the 5:

r^{2} = 5 \rightarrow r = \sqrt{5}

And with the center of the circumference and the radius already calculated, we can well proceed to draw our circumference:


Figure 3. Circumference (x + 1)^{2} + (y - 1)^{2} = 5

Elements and lines of the circumference

In this section we will mention the parts that make up a circumference and we will also mention the name given to the lines or segments that cut the circumference.

Circumference radius

The radius is a segment that joins the center with any point on the circumference.

Figure 4. Radius of the circumference

We can see that the radius of the circumference is the segment:


Circumference diameter

The diameter is the segment that joins two points on the circumference passing through the center.

Figure 5. Circumference diameter

Diameter \overline{WS} passing through the center C.

Arc of circumference

The arc of the circumference is a part of the same circumference delimited by any two points.

Figure 6. Arc of the circumference

The arc is named as:

\text{Arc } \ \widehat{WR}

String of the circumference

The chord is a segment that joins two points on the circumference.

Figure 7. Chord of the circumference

As can be seen in the previous figure, the circumference has a:

\text{String } \ \overline{WR}

Line tangent to the circumference

It is called a tangent line to the circumference when the line touches the circumference at a point, receiving the name of a point of tangency.

Figure 8. Line tangent to the circumference

W is the point of tangency.

Secant line to circumference

The secant line is the line that crosses the circumference at two points, resulting that a part of the same line is chord of the circumference.

Figure 9. Secant line to the circumference

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