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Circles Class 10 Notes Maths Chapter 10

Introduction to Circles 

Circle: It is a closed two-dimensional geometrical figure, such that all points on the surface of a circle are equidistant from the point called the “centre”.
A circle is a locus of a point that moves in such a way that the distance from that point is always fixed.

Parts of a CircleParts of a Circle

  • Radius: The constant distance from the centre to the circumference (boundary) of the circle.
  • Secant: A secant is a line that crosses a curve at two or more separate locations. A secant intersects a circle at exactly two locations in the case of a circle. In the figure, the line FG intersects the circle at two points P and Q. FG is the secant of the circle.SecantSecant
  • Chord: Any line segment joining the two points on the circumference of the circle. In the above figure, PQ is a chord.
  • Diameter: The longest distance between the two points on the circumference of the circle. It is the longest chord.Circles Class 10 Notes Maths Chapter 10Here, AO is the radius of the circle and AB is the diameter of the circle.
  • 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 of a Circle
    Tangent of a Circle
  • Non-intersecting lines: These are made up of two or more lines that do not intersect. The circle and the line AB have no common point. 

    Circles Class 10 Notes Maths Chapter 10

    - Lines that do not intersect can never meet.
    - The parallel lines are another name for them.
    - They stay at the same distance from one another at all times. 

Tangent to a Circle

 A tangent to a circle is a line that intersects the circle at only one point.

(i) There is only one tangent at a point of the circle.
Circles Class 10 Notes Maths Chapter 10

(ii) The tangent to a circle is a special case of the secant when the two endpoints of its corresponding chord coincide.
Circles Class 10 Notes Maths Chapter 10(iii) The common point of the tangent and the circle is called the point of contact and the tangent is said to touch the circle at that point.
Circles Class 10 Notes Maths Chapter 10

Question for Chapter Notes: Circles
Try yourself:
What is the definition of a circle?
View Solution

Theorem 1

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

Circles Class 10 Notes Maths Chapter 10

According to the theorem, O is the centre and OP⊥XY.

Circles Class 10 Notes Maths Chapter 10

Circles Class 10 Notes Maths Chapter 10

Note:
1. By the theorem above, we can also conclude that at any point on a circle, there can be one and only one tangent.
2. The line containing the radius through the point of contact is also sometimes called the ‘normal’ to the circle at the point.

Example 1: In the given figure, O is the centre of a circle, AB is a chord and AT is the tangent at A. If ∠AOB = 100°, then calculate ∠BAT.
Circles Class 10 Notes Maths Chapter 10

Solution:

Circles Class 10 Notes Maths Chapter 10

∠1 = ∠2

∠1 + ∠2 + 100° = 180°

∠1 + ∠1 = 80°

⇒ 2∠1 = 80°

⇒ ∠1 = 40°

∠1 + ∠BAT = 90°

∠BAT = 90° – 40° = 50°

Question for Chapter Notes: Circles
Try yourself:What is the relationship between the tangent to a circle at any point and the radius of the circle that passes through the point of contact?
View Solution

Number of Tangents from a Point on a Circle

1. There is no tangent to a circle passing through a point lying inside the circle.

If a point lies inside a circle, any line passing through that point will intersect the circle at two points and is called a secant. Therefore, it is not possible to draw a tangent to a circle that passes through a point inside the circle.

Circles Class 10 Notes Maths Chapter 10

2. There is one and only one tangent to a circle passing through a point lying on the circle.

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

Circles Class 10 Notes Maths Chapter 10

3. There are exactly two tangents to a circle through a point lying outside the circle

If a point is located outside of a circle, then there exist exactly two tangents that can be drawn to the circle passing through the point.

Circles Class 10 Notes Maths Chapter 10

Length of a Tangent

To define the length of a tangent from a point (P) to a circle, we measure the distance from the external point P to the point of tangency "I" on the circle. This distance is known as the tangent length (PI).

Circles Class 10 Notes Maths Chapter 10

Theorem 2

The lengths of tangents drawn from an external point to a circle are equal.

Circles Class 10 Notes Maths Chapter 10

Circles Class 10 Notes Maths Chapter 10

Circles Class 10 Notes Maths Chapter 10

Note:

1. The theorem can also be proved by using the Pythagoras Theorem as follows:
PA2 = OP2 – OA2 = OP2 – OB2 = PB2 (As OA = OB) which gives PA = PB.
2. Note also that ∠ OPA = ∠ OPB. Therefore, OP is the angle bisector of ∠ APB,i.e., the centre lies on the bisector of the angle between the two tangents.

Example 2: In the given figure, AB and AC are tangents to the circle with centre o such that ∠BAC = 40°. Then calculate ∠BOC.
Circles Class 10 Notes Maths Chapter 10

Solution:

Circles Class 10 Notes Maths Chapter 10

AB and AC are tangents
∴ ∠ABO = ∠ACO = 90°
In ABOC,
∠ABO + ∠ACO + ∠BAC + ∠BOC = 360°
90° + 90° + 40° + ∠BOC = 360°
∠BOC = 360 – 220° = 140°

Question for Chapter Notes: Circles
Try yourself:What is the relationship between the tangents drawn from an external point to a circle?
View Solution

Example 3: In the given figure, PA and PB are tangents to the circle with centre O. If ∠APB = 60°, then calculate ∠OAB.
Circles Class 10 Notes Maths Chapter 10

Solution:

Circles Class 10 Notes Maths Chapter 10

∠1 = ∠2
∠1 + ∠2 + ∠APB = 180°
∠1 + ∠1 + 60° = 180°
2∠1 = 180° – 60° = 120°
1

∠1 =  60°
∠1 + ∠OAB = 90°
60° +∠OAB = 90°
∠OAB = 90° – 60° = 30°

Example 4: In the given figure, O is the centre of a circle, PQ is a chord and PT is the tangent at P. If ∠POQ = 70°, then calculate ∠TP.
Circles Class 10 Notes Maths Chapter 10

Solution:
Circles Class 10 Notes Maths Chapter 10
∠1 = ∠2
∠1 + ∠2 + 70° = 180°
∠1 + ∠1 = 180° – 70°
2∠1 = 110° ⇒ ∠1 = 55°
∠1 + ∠TPQ = 90°
55° + ∠TPQ = 90°
⇒ ∠TPQ = 90° – 55° = 35°

Example 5: In the given figure, PQ R is a tangent at a point C to a circle with centre O. If AB is a diameter and ∠CAB = 30°. Find ∠PCA.
Circles Class 10 Notes Maths Chapter 10

Solution:
∠ACB = 90° …[Angle in the semi-circle
In ∆ABC,
∠CAB + ∠ACB + ∠CBA = 180°
30 + 90° + ∠CBA = 180°
∠CBA = 180° – 30° – 90° = 60°
∠PCA = ∠CBA …[Angle in the alternate segment
∴ ∠PCA = 60°

Summary

  • A circle is a two-dimensional figure formed by a set of all points equidistant from a fixed point in the same plane.
  • The fixed point is called the centre of the circle, and the fixed distance from the centre is known as the radius.
  • A secant is a line that intersects a circle at two distinct points.
  • A tangent to a circle is a line that touches the circle at exactly one point.
  • The tangent at any point on a circle is perpendicular to the radius drawn to the point of contact.
  • The lengths of tangents drawn from an external point to a circle are equal.

The document Circles Class 10 Notes Maths Chapter 10 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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FAQs on Circles Class 10 Notes Maths Chapter 10

1. How many tangents can be drawn from a point outside a circle?
Ans. A maximum of two tangents can be drawn from a point outside a circle to the circle.
2. Can a circle have a tangent at more than one point?
Ans. No, a circle can have only one tangent at a time from a given point on the circumference.
3. Is the tangent to a circle always perpendicular to the radius at the point of contact?
Ans. Yes, the tangent to a circle is always perpendicular to the radius at the point of contact.
4. How does the number of tangents from a point on a circle change based on the position of the point?
Ans. If the point is outside the circle, two tangents can be drawn. If the point is on the circle, only one tangent can be drawn. If the point is inside the circle, no tangents can be drawn.
5. Can a line passing through the center of a circle be considered as a tangent to the circle?
Ans. No, a line passing through the center of a circle is not considered as a tangent to the circle as it does not touch the circle at only one point.
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