Class 10 Maths Chapter 10: Circles

Circles is Chapter 10 of NCERT Class 10 Maths. Building on what you know about circles from Class 9, this chapter focuses on tangents: what a tangent is, how many tangents can be drawn from a point, and two important theorems about them. Watch the three video lessons below, then revise the theorems, results and solved examples given here.

What you will learn

• The difference between a secant and a tangent to a circle
• How many tangents can be drawn from a point inside, on or outside a circle
• The theorem that a tangent is perpendicular to the radius at the point of contact
• The theorem that tangents from an external point are equal in length
• Applying these results to find lengths and angles using Pythagoras theorem

Video Lessons

Circles - Part 1

Circles - Part 2

Circles - Part 3

Key Concepts and Formulas

Secant and Tangent

A line and a circle can meet in three ways. The line may not touch the circle at all, it may cut the circle at two points, or it may touch it at exactly one point.

• Non-intersecting line: has no common point with the circle.
• Secant: a line that meets the circle at two distinct points.
• Tangent: a line that meets the circle at exactly one point, called the point of contact.
• A tangent can be thought of as the limiting position of a secant when its two points of intersection come together.
• At any point on a circle there is one and only one tangent.

Number of Tangents from a Point

The position of the point decides how many tangents you can draw.

• Point inside the circle: no tangent can be drawn.
• Point on the circle: exactly one tangent.
• Point outside the circle: exactly two tangents.
• The length of a tangent is the length of the segment from the external point to the point of contact.

Theorem: Tangent and Radius

This is the first key theorem of the chapter and is often asked as a proof question in the CBSE board exam.

Tangent at P ⊥ radius OP
Tangent length = √(OP² − r²)
where OP = distance of external point from centre O
and r = radius

• Statement: the tangent at any point of a circle is perpendicular to the radius through the point of contact.
• Idea of proof: every point on the tangent other than the point of contact lies outside the circle, so the radius is the shortest distance from the centre to the tangent, and the shortest segment is the perpendicular.
• The line through the point of contact, perpendicular to the tangent, passes through the centre.

Theorem: Equal Tangents

The second theorem is proved using congruent triangles (RHS rule).

PA = PB  (tangents from external point P)
∠APB + ∠AOB = 180°

• Statement: the lengths of the two tangents drawn from an external point to a circle are equal.
• Proof outline: for tangents PA and PB, triangles OAP and OBP have OA = OB (radii), OP common, and ∠OAP = ∠OBP = 90°, so they are congruent by RHS and PA = PB.
• The centre lies on the bisector of the angle between the two tangents.
• The angle between the two tangents and the angle they subtend at the centre are supplementary.

Useful Results for Problems

These results follow directly from the two theorems and appear often in short-answer questions.

Quadrilateral ABCD circumscribing a circle:
AB + CD = AD + BC

• For concentric circles, a chord of the larger circle that touches the smaller circle is bisected at the point of contact.
• Tangents drawn at the ends of a diameter are parallel to each other.

Solved Examples

Example 1: A point P is 17 cm from the centre of a circle of radius 8 cm. Find the length of the tangent from P.

Tangent length = √(OP² − r²)
= √(17² − 8²)
= √(289 − 64)
= √225
∴ Tangent length = 15 cm

Example 2: The tangent from a point 29 cm away from the centre of a circle has length 20 cm. Find the radius.

r = √(OP² − PT²)
= √(29² − 20²)
= √(841 − 400)
= √441
∴ Radius = 21 cm

Example 3: PA and PB are tangents from P to a circle with centre O, and ∠APB = 70°. Find ∠AOB and ∠OAB.

∠AOB = 180° − ∠APB
∠AOB = 180° − 70° = 110°
OA = OB, so ∠OAB = ∠OBA
∠OAB = (180° − 110°) ÷ 2
∴ ∠AOB = 110°, ∠OAB = 35°

Example 4: Two concentric circles have radii 10 cm and 6 cm. Find the length of the chord of the larger circle that touches the smaller circle.

Half chord = √(10² − 6²)
= √(100 − 36)
= √64 = 8 cm
Chord = 2 × 8
∴ Chord length = 16 cm

Example 5: A quadrilateral ABCD circumscribes a circle with AB = 9 cm, BC = 11 cm and CD = 8 cm. Find AD.

AB + CD = AD + BC
9 + 8 = AD + 11
AD = 17 − 11
∴ AD = 6 cm

Example 6: Tangents PA and PB are drawn from P with PA = 7 cm and ∠APB = 60°. Find AB.

PA = PB = 7 cm
∠PAB = ∠PBA = (180° − 60°) ÷ 2 = 60°
All angles 60°, so △PAB is equilateral
∴ AB = 7 cm

Common Mistakes to Avoid

• Drawing a figure without marking the 90° angle between radius and tangent. This right angle is the key to almost every numerical question.
• Using ∠APB + ∠AOB = 90° instead of 180°. Remember the quadrilateral OAPB has two right angles, so the other two angles add up to 180°.
• Writing incomplete proofs. For the theorems, always state what is given, what is to be proved, the construction and the reasons for each step.
• Mixing up which side is the hypotenuse. The line joining the centre to the external point (OP) is always the hypotenuse.

Quick Revision

Radius ⊥ tangent at point of contact
PA = PB
∠APB + ∠AOB = 180°
Tangent length = √(OP² − r²)
AB + CD = AD + BC (circumscribed quadrilateral)

• Secant: two common points; tangent: exactly one common point
• Inside: 0 tangents, on: 1 tangent, outside: 2 tangents

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