Circles Class 10 Notes: Chapter 10

CBSE Class 10 Maths Circles Notes:-Download PDF Here

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

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.

Circles for class 10 -1

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

Tangent

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

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Tangent

Secant

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

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Secant

Tangent as a special case of Secant

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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.

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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”.

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Tangent and radius

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.

AB is a secant drawn
through the point S

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

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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

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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.

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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.

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Tangents to a circle from an external point

PT1=PT2

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