CBSE Grade 10 Maths Chapter 10 - Circles


What is a Circle ?

  • A figure made by all the points which are at the same distance from a fixed point is called a Circle.
  • A circle has following parts:

1. Center

  • The fixed point is known as the centre of the circle.

2. Radius

  • The distance from any point on the circle to the fixed point is the radius. 
  • Any line segment which joins the centre and any point on the circle is known as the Radius.

3. Chord

  • Any line segment made by joining any two points on the boundary of the circle is called Chord.

4. Diameter

  • Diameter is two times or twice the radius.
  • It is the longest chord on the circle which passes through the centre.
  • All the diameters have the same length.

5. Circumference

  • The length of the boundary of the circle is called the circumference of the circle.

6. Arc

An arc is the part of the circle joining two points on the circumference of the circle.

7. Sector

An area made by an arc and two radii of the circle, by joining the centre to the endpoints of the arc is called Sector.

8. Segment

An area made by a chord and an arc of the circle is called Segment.

The Relation between a Circle and a Line in a Plane

There could be three situations when there are a line and a circle.


The Relation between a Circle and a Line in a Plane

1. Non-intersecting Line

  • When a line and a circle have no common point then it is called a Non-intersecting line with respect to the circle. 
  • In above figure , line 'l' is a non-intersecting line.

2. Secant

  • When a line intersects a circle in such a way that there are two common points then that line is called Secant.
  • It cuts the circle at two points, forming a chord of the circle.
  • In above circle, Secant is intersecting Circle at points A an B.

3. Tangent

  • When a line touches the circle in such a way that they have only one common point i.e.the line touches the circle.at one point ,then that line is called a Tangent.
  • That common point, where line touches the circle is called the point of contact.
  • In the above circle, tangent is touching the circle at Point P. So Point P is point of contact.

Tangent to a Circle

  • 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.
  • Theorem : Tangent perpendicular to the radius at the point of contact
     
All the tangents of a circle are perpendicular to the radius through the point of contact of that tangent.


OP is the radius of the circle and Q is any point on the line XY . XY is the tangent to the circle. As OP is the shortest line of all the distances of the point O to the points on XY. So OP is perpendicular to XY. Hence, OP⊥ XY

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

  • 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.

     

Example

Find the radius of the circle in the given figure, if the length of the tangent from point A which is 5 cm away from center is 4 cm.

Solution


Number of Tangents from a Point on a Circle

1. There could be only one tangent at one point of contact.


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

2. Tangent could not be drawn from any point inside the circle.

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.

3. There could be only two tangents to a circle from any point outside the circle.


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

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.



Here, two tangents are drawn from the external point C.

As the tangent is perpendicular to the radius, it formed the right angle triangle.

So ∆AOC and ∆BOC are congruent right angle triangle.

Hence AC = BC.

Example

If two tangents PA and PB are drawn to a circle from a point P with centre O and OP is equal to the diameter of the circle then show that triangle APB is an equilateral triangle.


Two tangents PA and PB are drawn



Example

Find the length of AB in the given circle, which is the chord in the outer circle and tangent to the inner circle. The radius of the inner and outer circle is 6 cm and 10 cm respectively.


Solution

Given:


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