# Circles Class 10 Notes: Chapter 10

## CBSE Class 10 Maths Circles Notes:-

A brief introduction to circles for class 10 is provided here. Get the complete description provided here to learn about the concept of the circle. Also, learn how to draw a tangent to the circle with various theorems and examples.

## Introduction to Circles

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### Circle and line in a plane

For a circle and a line on a plane, there can be three possibilities.

i) they can be non-intersecting

ii) they can have a single common point: in this case, the line touches the circle.

ii) they can have two common points: in this case, the line cuts the circle.

(i) Non intersecting   (ii) Touching  (iii) Intersecting

### Tangent

A tangent to a circle is a line that touches the circle at exactly one point. For every point on the circle, there is a unique tangent passing through it.

Tangent

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### Secant

A secant to a circle is a line that has two points in common with the circle. It cuts the circle at two points, forming a chord of the circle.

Secant

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### Tangent as a special case of Secant

Tangent as a special case of Secant

The tangent to a circle can be seen as a special case of the secant when the two endpoints of its corresponding chord coincide.

### Two parallel tangents at most for a given secant

For every given secant of a circle, there are exactly two tangents which are parallel to it and touches the circle at two diametrically opposite points.

Parallel tangents

## Theorems

### Tangent perpendicular to the radius at the point of contact

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

Here, O is the centre and OPXY.

### The number of tangents drawn from a given point

i) If the point is in an interior region of the circle, any line through that point will be a secant. So, no tangent can be drawn to a circle which passes through a point that lies inside it.

No tangent can be drawn to a circle from a point inside it

ii) When a point of tangency lies on the circle, there is exactly one tangent to a circle that passes through it.

A tangent passing through a point lying on the circle

iii) When the point lies outside of the circle, there are accurately two tangents to a circle through it

Tangents to a circle from an external point

### Length of a tangent

The length of the tangent from the point (Say P) to the circle is defined as the segment of the tangent from the external point P to the point of tangency I with the circle. In this case, PI is the tangent length.

### Lengths of tangents drawn from an external point

Theorem: Two tangents are of equal length when the tangent is drawn from an external point to a circle.

Tangents to a circle from an external point

PT1=PT2

Thus, the two important theorems in Class 10 Maths Chapter 10 Circles are:
Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal.
Interesting facts about Circles and its properties are listed below:

• In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.
• The tangents drawn at the ends of a diameter of a circle are parallel.
• The perpendicular at the point of contact to the tangent to a circle passes through the centre.
• The angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.
• The parallelogram circumscribing a circle is a rhombus.
• The opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

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