Circles: Definition, Properties, Parts, Theorems

A Circle is a curved 2-D figure formed by a set of points. A circle is measured in terms of its radius. In geometry, it is a special kind of ellipse where eccentricity is ‘0’ and two foci are coincident. 

In our daily life, we often come across such objects which have round shapes such as dials of many clocks, bangles, wheels of a vehicle, finger rings, coins, keyrings and so on. In a clock, we notice that the tip of the second’s hand moves in a round path. The path and the path by the tip of the second’s hand is a circle (or circular in motion). 

Key Terms: Circle, Center, Radius, Diameter, Tangent, Chord, Arc, Sector, Secant

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Definition

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A circle is the collection of all the points in a plane where, each of which is at a constant distance from a fixed point in that particular plane. In other way, we can say that a circle refers to the path of a point which moves in a plane in such a way that it remains at a constant distance from a fixed point in a plane .

The fixed point is the centre of the circle whereas, the constant distance is called the radius of the circle. The radius of a circle is always positive and all radii of a circle are equal.

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Terms Related to a Circle

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Circle: A circle is the combination of points P in a plane that is equidistant from a fixed point.
Centre: The fixed point from where the circle is drawn.

Center of the circle

Radius: The equidistant line drawn from the centre is termed the radius.

Radius of the circle

Chord: A line segment connecting any two points on a circle is termed a chord of the circle.

Chord of the circle

Diameter: A chord passing through the centre of the circle is called the diameter of the circle. It's the longest chord of the circle and all diameters have equal length.

Diameter of the circle

Arc of a circle: An arc of a circle indicates the continuous part of a circle. The arc of a circle is denoted by “?” this symbol. The arc of a circle is again categorised into three parts: Minor and Major arc, Semicircle and Circumference.

Arc of a circle

Sector of a circle: The area of the plane region enclosed by an arc of a circle and its two bounding radii is known as the sector of the circle.

Sector of a circle

Circle and line during a plane: There may be three possible outcomes for a circle and a line on a Plane,

1. They'll be non-intersecting.
2. They'll have a line touching the circle with one common point. This line is taken into account to be a tangent.
3. They'll have a line intersecting two common points of the circle; this line is considered to be a secant.

Now, We can derive the definition of tangent and secant from this explanation.

Tangent- The straight line that meets the circle at one point or two is known as a tangent.

Through the point of contact, the tangent drawn to a circle is usually perpendicular to the radius.

Secant- A straight line intersecting the circle at two points is called a secant or an extended chord.

Length of a tangent- The length of the tangent drawn from a point, let's suppose X to the circle is named the segment of the tangent from the external point X to the point of tangency Y with the circle. In this case, XY is the tangent length.

Length of a tangent

Properties of a Tangent drawn to a circle

• Only one tangent can touch a circle at one point of the circle.
• A tangent can not pass through the circle as a tangent never crosses a circle.
• A tangent can never intersect the circle at two points.
• The line of tangency is perpendicular to the radius of a circle.
• The length of two tangents from a common external point is always equal to a circle.

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Theorems

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The theorem of tangent states that a line will be a tangent to a circle if and only if the line is perpendicular to the radius that is drawn to the point of tangency. 

Tangent is perpendicular to the radius at the point of contact

Theorem: This theorem proves that “the tangent drawn to the circle at any point is the perpendicular to the radius of the circle that passes through the point of contact”.

The number of tangents drawn from a given point

1. If the point is inside the circle, any line through that point is a secant. So, no tangent can be drawn to a circle that passes through a point that lies inside it.
2. There is exactly one tangent to a circle that passes through it if a point of tangency lies on the circle, 
3. There are accurately two tangents to a circle through it, when the point lies outside of the circle.

Tangent is perpendicular to the radius at the point of contact

Lengths of tangents drawn from an external point

Theorem: The second theorem states that two tangents will be of equal length when the tangent is drawn from an external point to a circle. The tangent segment refers to the line that joins the external point and the point of tangency. 

Lengths of tangents drawn from an external point

Tangent as a special case of Secant

When the two endpoints of its corresponding chord coincide, the tangent to a circle may be considered as a special case of the secant. 

Moreover, a circle can possess two parallel tangents at most for a given secant.

There are exactly two tangents that are parallel to that and intersect the circle at two diametrically opposite points, for every given secant of a circle.

Tangent as a special case of Secant

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